🧠:<think>Okay, so I have this problem about Dr. Lexi Verba analyzing language evolution and cultural influence. It's divided into two parts. Let me try to tackle them one by one.Starting with part 1: We have a function h(t, x) which is the product of f(t) and g(x). f(t) is given as e^{-kt²}, and g(x) is a sine function: a sin(bx + c) + d. The task is to find the critical points of h(t, x) with respect to t for a fixed x.Hmm, critical points occur where the derivative is zero or undefined. Since h is a function of both t and x, but x is fixed here, I just need to take the partial derivative of h with respect to t and set it equal to zero.So, h(t, x) = f(t) * g(x) = e^{-kt²} * [a sin(bx + c) + d]. Since x is fixed, g(x) is just a constant with respect to t. Let me denote g(x) as a constant, say, G. So, h(t) = G * e^{-kt²}.To find critical points, take the derivative dh/dt and set it to zero.dh/dt = G * d/dt [e^{-kt²}] = G * e^{-kt²} * (-2kt).Set this equal to zero:G * e^{-kt²} * (-2kt) = 0.Now, e^{-kt²} is never zero, G is a constant (could be zero, but if G is zero, then h(t) is zero everywhere, which is trivial). So, assuming G ≠ 0, then we have:-2kt * e^{-kt²} = 0.Again, e^{-kt²} is never zero, so we can divide both sides by it, getting:-2kt = 0.Solving for t: t = 0.So, the only critical point is at t = 0.Wait, but is that the only critical point? Let me think. The function h(t) is G * e^{-kt²}, which is a Gaussian-like curve centered at t=0. Its derivative is proportional to -2kt * e^{-kt²}, which is zero only at t=0. So yes, only one critical point at t=0.But hold on, is t=0 a maximum or a minimum? Let's check the second derivative or just analyze the behavior.Since k is positive, as t increases, e^{-kt²} decreases. So, the function h(t) has a maximum at t=0. So, t=0 is a maximum point.But the question just asks for critical points, so t=0 is the only critical point.Wait, but is that the case for all x? Since g(x) is a sin function, which can be positive or negative. If G is negative, does that affect the critical point?No, because when we set dh/dt = 0, we get t=0 regardless of G's sign. Because even if G is negative, the derivative is still proportional to -2kt, so setting it to zero gives t=0. So, regardless of x, the critical point is at t=0.So, conclusion: The only critical point is at t=0.Moving on to part 2: Dr. Verba models the variance V(t) as the integral from 0 to T of f²(t) * g(x) dx. So, V(t) = ∫₀^T [e^{-kt²}]² * [a sin(bx + c) + d] dx.Simplify that: [e^{-kt²}]² = e^{-2kt²}, so V(t) = e^{-2kt²} * ∫₀^T [a sin(bx + c) + d] dx.Wait, hold on, is V(t) dependent on t? Because f(t) is e^{-kt²}, so f²(t) is e^{-2kt²}, which is a function of t. Then, V(t) is e^{-2kt²} multiplied by the integral of g(x) over x from 0 to T.But wait, the integral ∫₀^T g(x) dx is just a constant with respect to t, right? Because it's integrating over x, and t is a variable here.Wait, but hold on, is V(t) a function of t? Because f²(t) is e^{-2kt²}, which is multiplied by the integral of g(x) over x. So, V(t) is proportional to e^{-2kt²} times a constant.So, V(t) = C * e^{-2kt²}, where C = ∫₀^T [a sin(bx + c) + d] dx.So, to find the maximum of V(t) over 0 ≤ t ≤ T, we can analyze this function.Since C is a constant, the maximum of V(t) occurs where e^{-2kt²} is maximized. The exponential function e^{-2kt²} is a decreasing function for t > 0 because the exponent is negative and becomes more negative as t increases.Therefore, e^{-2kt²} is maximum at t=0, and it decreases as t increases. Hence, V(t) reaches its maximum at t=0.But wait, is that always the case? What if C is negative? Because if C is negative, then V(t) would be a negative number, and its maximum would be at the smallest magnitude, which is still t=0.Wait, but V(t) is supposed to represent variance, which should be non-negative. So, maybe C must be positive? Let me check.C = ∫₀^T [a sin(bx + c) + d] dx.So, the integral of a sine function over an interval plus d times the interval.The integral of sin(bx + c) over 0 to T is [-cos(bx + c)/b] from 0 to T, which is [-cos(bT + c) + cos(c)] / b.So, C = a * [(-cos(bT + c) + cos(c)) / b] + d * T.So, C is a combination of these terms. Depending on the values of a, b, c, d, T, C could be positive or negative.But since V(t) is a variance, it should be non-negative. So, perhaps C is non-negative? Or maybe the model assumes that C is positive.But regardless, for the purpose of finding the maximum of V(t), which is C * e^{-2kt²}, the maximum occurs at t=0 because e^{-2kt²} is maximum there.So, the maximum of V(t) occurs at t=0.But the question says "determine the conditions under which V(t) reaches its maximum value within the interval 0 ≤ t ≤ T."So, the maximum is at t=0, but we need to ensure that t=0 is within the interval, which it is, since 0 ≤ t ≤ T.But maybe there are cases where the maximum could be elsewhere? Wait, if C is zero, then V(t) is zero everywhere. If C is positive, it's maximum at t=0. If C is negative, then V(t) is negative, and its maximum (i.e., least negative) is at t=0.But since variance should be non-negative, perhaps C must be non-negative. So, the condition is that C ≥ 0.So, the maximum of V(t) is at t=0 provided that C ≥ 0.Therefore, the condition is that ∫₀^T [a sin(bx + c) + d] dx ≥ 0.So, to write that out:∫₀^T [a sin(bx + c) + d] dx ≥ 0.Which is:a * [(-cos(bT + c) + cos(c)) / b] + d * T ≥ 0.So, that's the condition.Alternatively, if we don't simplify, it's just the integral of g(x) over x from 0 to T must be non-negative.Therefore, the maximum of V(t) occurs at t=0 provided that ∫₀^T g(x) dx ≥ 0.So, summarizing:1. The critical point of h(t, x) with respect to t for fixed x is at t=0.2. The variance V(t) reaches its maximum at t=0 provided that the integral of g(x) over [0, T] is non-negative.I think that's it. Let me just double-check.For part 1, h(t, x) = f(t)g(x). Since g(x) is treated as a constant when taking derivative w.r. to t, the derivative is f’(t)g(x). Setting to zero, f’(t)=0, which occurs at t=0 because f(t)=e^{-kt²} has its maximum at t=0.For part 2, V(t) is proportional to e^{-2kt²}, which is a decreasing function, so maximum at t=0, but only if the proportionality constant is non-negative. If the integral of g(x) is negative, then V(t) would be negative, but variance can't be negative, so the model likely assumes the integral is non-negative.Yeah, that makes sense.Final Answer1. The critical point is at boxed{0}.2. The maximum variance occurs at ( t = 0 ) provided that ( int_{0}^{T} g(x) , dx geq 0 ). Thus, the condition is boxed{int_{0}^{T} g(x) , dx geq 0}.</think>

🧠:<think>First, I need to calculate the total force exerted by the floodwater on the window. The window is 2 meters wide and 3 meters tall, and the water has risen to a height of 1.5 meters inside the house.I'll start by determining the hydrostatic pressure at the bottom of the window using the formula ( P = rho g h ), where ( rho ) is the density of water (1000 kg/m³), ( g ) is the acceleration due to gravity (9.8 m/s²), and ( h ) is the height of the water (1.5 meters). This will give me the pressure at the deepest point of the window.Next, I'll calculate the average pressure on the window. Since the pressure increases with depth, the average pressure is half of the maximum pressure at the bottom. This average pressure will be used to find the total force exerted on the window by multiplying it by the window's area.Moving on to the second part, I need to determine the total cost of replacing the living room flooring. The living room measures 8 meters by 10 meters, so I'll first calculate the area. The cost per square meter for the flooring is 50, and there's an additional 20 per square meter for labor. I'll multiply the total area by the sum of these costs to find the overall cost.Finally, since the homeowner receives insurance coverage of 75% of the total cost, I'll calculate the insured amount and subtract it from the total cost to determine how much the homeowner needs to pay out of pocket.</think>

🧠:<think>Alright, so I have this problem where a dietitian is trying to design a nutritional plan for a patient recovering from surgery. The absorption rates of two nutrients, N₁ and N₂, are modeled by a system of differential equations. I need to solve two sub-problems: first, derive the general solution for the system, and second, find the conditions on the constants so that the nutrients reach a steady state. Okay, starting with the first sub-problem. The system is given by:1. dN₁/dt = aN₁ - bN₂ + c2. dN₂/dt = -dN₁ + eN₂ + fAll constants are non-zero, and the system is stable. So, I need to solve this system of linear differential equations. I remember that for systems like this, we can write them in matrix form and find eigenvalues and eigenvectors to solve them. Alternatively, we can use methods like substitution or elimination to solve them step by step.Let me write the system in matrix form:d/dt [N₁; N₂] = [a -b; -d e] [N₁; N₂] + [c; f]So, it's a linear system with constant coefficients and a constant forcing function. The general solution will be the sum of the homogeneous solution and a particular solution.First, let's find the homogeneous solution. The homogeneous system is:dN₁/dt = aN₁ - bN₂dN₂/dt = -dN₁ + eN₂To solve this, I can write the characteristic equation. The matrix is:| a - λ -b || -d e - λ |The determinant of this matrix should be zero for eigenvalues λ.So, the characteristic equation is (a - λ)(e - λ) - (-b)(-d) = 0Calculating that:(a - λ)(e - λ) - bd = 0Expanding (a - λ)(e - λ):ae - aλ - eλ + λ² - bd = 0So, λ² - (a + e)λ + (ae - bd) = 0This is a quadratic equation in λ. The solutions will be:λ = [(a + e) ± sqrt((a + e)² - 4(ae - bd))]/2Simplify the discriminant:Δ = (a + e)² - 4(ae - bd) = a² + 2ae + e² - 4ae + 4bd = a² - 2ae + e² + 4bdWhich is (a - e)² + 4bdSince the system is stable, the real parts of the eigenvalues must be negative. So, the eigenvalues should have negative real parts, which would mean that the system converges to the equilibrium.Now, assuming that the eigenvalues are real and distinct, or complex conjugates. But since the discriminant is (a - e)² + 4bd, which is always positive because it's a square plus a positive term (since b, d are non-zero, but wait, are they positive? The problem says constants are non-zero, but doesn't specify sign. Hmm. Maybe I need to proceed without assuming the nature of eigenvalues.But for the general solution, regardless of the eigenvalues, we can express it in terms of eigenvalues and eigenvectors.But perhaps it's easier to find the particular solution and then the homogeneous solution.Alternatively, since the system is linear, I can try to solve it using substitution.Let me try to express N₂ from the first equation and substitute into the second.From equation 1:dN₁/dt = aN₁ - bN₂ + cSo, rearranged:bN₂ = aN₁ - dN₁/dt + cThus,N₂ = (a/b)N₁ - (1/b)dN₁/dt + c/bNow, plug this into equation 2:dN₂/dt = -dN₁ + eN₂ + fFirst, compute dN₂/dt:d/dt [ (a/b)N₁ - (1/b)dN₁/dt + c/b ] = (a/b)dN₁/dt - (1/b)d²N₁/dt² + 0So, dN₂/dt = (a/b)dN₁/dt - (1/b)d²N₁/dt²Now, substitute into equation 2:(a/b)dN₁/dt - (1/b)d²N₁/dt² = -dN₁ + e[(a/b)N₁ - (1/b)dN₁/dt + c/b] + fLet me expand the right-hand side:= -dN₁ + e(a/b)N₁ - e(1/b)dN₁/dt + e(c/b) + fSo, putting it all together:(a/b)dN₁/dt - (1/b)d²N₁/dt² = -dN₁ + (ea/b)N₁ - (e/b)dN₁/dt + (ec)/b + fNow, multiply both sides by b to eliminate denominators:a dN₁/dt - d²N₁/dt² = -b dN₁ + ea N₁ - e dN₁/dt + ec + bfBring all terms to the left-hand side:a dN₁/dt - d²N₁/dt² + b dN₁ - ea N₁ + e dN₁/dt - ec - bf = 0Combine like terms:(-d²N₁/dt²) + (a + e) dN₁/dt + (b - ea) N₁ - ec - bf = 0So, we have a second-order linear differential equation for N₁:d²N₁/dt² - (a + e) dN₁/dt - (b - ea) N₁ + ec + bf = 0Wait, let me check the signs:From the previous step:- d²N₁/dt² + (a + e) dN₁/dt + (b - ea) N₁ - ec - bf = 0So, multiplying both sides by -1:d²N₁/dt² - (a + e) dN₁/dt - (b - ea) N₁ + ec + bf = 0Yes, that's correct.So, the equation is:d²N₁/dt² - (a + e) dN₁/dt - (b - ea) N₁ + (ec + bf) = 0This is a linear nonhomogeneous ODE. To solve this, we can find the homogeneous solution and then a particular solution.First, homogeneous equation:d²N₁/dt² - (a + e) dN₁/dt - (b - ea) N₁ = 0Characteristic equation:r² - (a + e) r - (b - ea) = 0Solutions:r = [ (a + e) ± sqrt( (a + e)^2 + 4(b - ea) ) ] / 2Simplify discriminant:Δ = (a + e)^2 + 4(b - ea) = a² + 2ae + e² + 4b - 4ea = a² - 2ae + e² + 4bWhich is similar to the earlier discriminant but not exactly the same. Wait, earlier we had (a - e)^2 + 4bd, but here it's (a - e)^2 + 4b. Hmm, because in the substitution, we might have lost some terms. Wait, no, because in the substitution, we had to express N₂ in terms of N₁ and its derivative, which might have changed the coefficients.But regardless, the characteristic equation is quadratic, so we'll have two roots, say r₁ and r₂.Assuming distinct real roots, the homogeneous solution is N₁_h = C₁ e^{r₁ t} + C₂ e^{r₂ t}If the roots are complex, it'll be in terms of sines and cosines. But since the system is stable, the real parts of the eigenvalues must be negative, so the solutions will decay to zero.Now, for the particular solution, since the nonhomogeneous term is a constant (ec + bf), we can assume a constant particular solution N₁_p = K.Plugging into the ODE:0 - (a + e) * 0 - (b - ea) K + ec + bf = 0So:- (b - ea) K + ec + bf = 0Solving for K:K = (ec + bf)/(b - ea)So, the general solution for N₁ is:N₁(t) = C₁ e^{r₁ t} + C₂ e^{r₂ t} + (ec + bf)/(b - ea)Similarly, once we have N₁, we can find N₂ using the expression we derived earlier:N₂ = (a/b)N₁ - (1/b)dN₁/dt + c/bSo, plug in N₁(t):N₂(t) = (a/b)[C₁ e^{r₁ t} + C₂ e^{r₂ t} + K] - (1/b)[C₁ r₁ e^{r₁ t} + C₂ r₂ e^{r₂ t}] + c/bSimplify:N₂(t) = (a/b)C₁ e^{r₁ t} + (a/b)C₂ e^{r₂ t} + (a/b)K - (C₁ r₁ / b) e^{r₁ t} - (C₂ r₂ / b) e^{r₂ t} + c/bCombine like terms:= [ (a/b - r₁ / b) C₁ ] e^{r₁ t} + [ (a/b - r₂ / b) C₂ ] e^{r₂ t} + (a/b)K + c/bNow, let's compute (a/b - r / b) where r is r₁ or r₂.From the characteristic equation, r satisfies r² - (a + e) r - (b - ea) = 0So, r² = (a + e) r + (b - ea)Thus, r = (a + e) + (b - ea)/rWait, maybe another approach. Let's express (a - r)/b:(a - r)/b = (a - r)/bBut from the characteristic equation:r² = (a + e) r + (b - ea)So, r² - (a + e) r = b - eaDivide both sides by b:(r² - (a + e) r)/b = (b - ea)/bBut I'm not sure if that helps. Alternatively, maybe express (a - r)/b in terms of the equation.Alternatively, perhaps it's better to leave it as is.So, the general solution for N₂(t) is as above.Now, applying initial conditions to find C₁ and C₂.Given N₁(0) = N_{1,0} and N₂(0) = N_{2,0}Compute N₁(0):N₁(0) = C₁ + C₂ + K = N_{1,0}Similarly, compute N₂(0):N₂(0) = (a/b)C₁ + (a/b)C₂ + (a/b)K - (C₁ r₁ / b) - (C₂ r₂ / b) + c/b = N_{2,0}So, we have two equations:1. C₁ + C₂ = N_{1,0} - K2. (a/b)(C₁ + C₂) - (C₁ r₁ + C₂ r₂)/b + (a/b)K + c/b = N_{2,0}But from equation 1, C₁ + C₂ = N_{1,0} - K, so we can substitute into equation 2:(a/b)(N_{1,0} - K) - (C₁ r₁ + C₂ r₂)/b + (a/b)K + c/b = N_{2,0}Simplify:(a/b)N_{1,0} - (a/b)K - (C₁ r₁ + C₂ r₂)/b + (a/b)K + c/b = N_{2,0}The - (a/b)K and + (a/b)K cancel out:(a/b)N_{1,0} - (C₁ r₁ + C₂ r₂)/b + c/b = N_{2,0}Multiply both sides by b:a N_{1,0} - (C₁ r₁ + C₂ r₂) + c = b N_{2,0}Thus,C₁ r₁ + C₂ r₂ = a N_{1,0} + c - b N_{2,0}So, now we have a system of two equations:1. C₁ + C₂ = N_{1,0} - K2. C₁ r₁ + C₂ r₂ = a N_{1,0} + c - b N_{2,0}We can solve this system for C₁ and C₂.Let me write it as:C₁ + C₂ = S, where S = N_{1,0} - KC₁ r₁ + C₂ r₂ = T, where T = a N_{1,0} + c - b N_{2,0}We can solve for C₁ and C₂ using Cramer's rule or substitution.Express C₂ = S - C₁Substitute into the second equation:C₁ r₁ + (S - C₁) r₂ = TC₁ (r₁ - r₂) + S r₂ = TThus,C₁ = (T - S r₂)/(r₁ - r₂)Similarly,C₂ = (T - S r₁)/(r₂ - r₁)So, plugging back S and T:C₁ = [ (a N_{1,0} + c - b N_{2,0}) - (N_{1,0} - K) r₂ ] / (r₁ - r₂)C₂ = [ (a N_{1,0} + c - b N_{2,0}) - (N_{1,0} - K) r₁ ] / (r₂ - r₁)But K is (ec + bf)/(b - ea), so we can substitute that in.This gives us the constants C₁ and C₂ in terms of the initial conditions and the parameters.Therefore, the general solution is:N₁(t) = C₁ e^{r₁ t} + C₂ e^{r₂ t} + KN₂(t) = [ (a - r₁)/b C₁ ] e^{r₁ t} + [ (a - r₂)/b C₂ ] e^{r₂ t} + (a K + c)/bWait, let me check the expression for N₂(t). Earlier, I had:N₂(t) = [ (a/b - r₁ / b) C₁ ] e^{r₁ t} + [ (a/b - r₂ / b) C₂ ] e^{r₂ t} + (a/b)K + c/bWhich can be written as:N₂(t) = [ (a - r₁)/b C₁ ] e^{r₁ t} + [ (a - r₂)/b C₂ ] e^{r₂ t} + (a K + c)/bYes, that's correct.So, putting it all together, the general solution is expressed in terms of the eigenvalues r₁ and r₂, the constants C₁ and C₂ determined by initial conditions, and the particular solution K.Now, for the second sub-problem: finding the conditions on the constants so that the nutrients reach a steady state N₁* and N₂*.A steady state occurs when dN₁/dt = 0 and dN₂/dt = 0.So, setting the derivatives to zero:0 = a N₁* - b N₂* + c0 = -d N₁* + e N₂* + fThis gives a system of linear equations:a N₁* - b N₂* = -c-d N₁* + e N₂* = -fWe can write this as:a N₁* - b N₂* = -c-d N₁* + e N₂* = -fTo solve for N₁* and N₂*, we can use Cramer's rule or solve the system.Let me write the system as:a N₁* - b N₂* = -c-d N₁* + e N₂* = -fLet me write it in matrix form:[ a -b ] [N₁*] = [ -c ][ -d e ] [N₂*] [ -f ]The determinant of the coefficient matrix is:Δ = a e - (-b)(-d) = a e - b dFor a unique solution, Δ ≠ 0. So, the condition is a e ≠ b d.Assuming Δ ≠ 0, the solution is:N₁* = ( (-c) e - (-b)(-f) ) / Δ = (-c e - b f)/ΔN₂* = ( a (-f) - (-d)(-c) ) / Δ = (-a f - d c)/ΔWait, let me compute using Cramer's rule.For N₁*:Replace the first column with the constants:Δ₁ = | -c -b | -f eΔ₁ = (-c)(e) - (-b)(-f) = -c e - b fSimilarly, for N₂*:Replace the second column:Δ₂ = | a -c | -d -fΔ₂ = a (-f) - (-c)(-d) = -a f - c dThus,N₁* = Δ₁ / Δ = (-c e - b f)/(a e - b d)N₂* = Δ₂ / Δ = (-a f - c d)/(a e - b d)So, the steady state levels are:N₁* = (-c e - b f)/(a e - b d)N₂* = (-a f - c d)/(a e - b d)But we can write these as:N₁* = (c e + b f)/(b d - a e)N₂* = (a f + c d)/(b d - a e)Because multiplying numerator and denominator by -1.So, the conditions for the existence of a steady state are that the determinant Δ = a e - b d ≠ 0. So, a e ≠ b d.Additionally, for the system to reach the steady state, the homogeneous solutions must decay to zero, which requires that the real parts of the eigenvalues r₁ and r₂ are negative.From the characteristic equation for the homogeneous system:r² - (a + e) r - (b - ea) = 0The eigenvalues are r = [ (a + e) ± sqrt( (a + e)^2 + 4(b - ea) ) ] / 2For the real parts to be negative, the sum of the roots (which is (a + e)) must be negative, and the product of the roots (which is -(b - ea)) must be positive.Wait, the sum of the roots is (a + e), and the product is -(b - ea) = ea - b.So, for stability:1. (a + e) < 02. (ea - b) > 0But wait, the product of the roots is (ea - b). For the roots to have negative real parts, if they are real, both roots must be negative. If they are complex, the real part must be negative.But in our case, the discriminant is (a + e)^2 + 4(b - ea). Wait, earlier we had for the second-order equation, the discriminant was (a - e)^2 + 4b, but in the homogeneous system for N₁, it's (a + e)^2 + 4(b - ea). Hmm, perhaps I need to double-check.Wait, the characteristic equation for the second-order ODE was:r² - (a + e) r - (b - ea) = 0So, discriminant Δ = (a + e)^2 + 4(b - ea)Which is (a + e)^2 + 4b - 4ea = a² + 2ae + e² + 4b - 4ea = a² - 2ae + e² + 4bWhich is (a - e)^2 + 4bSo, the discriminant is always positive because it's a square plus 4b. Since b is non-zero, but we don't know its sign. If b is positive, then discriminant is definitely positive. If b is negative, then 4b could make it smaller, but (a - e)^2 is always non-negative, so as long as (a - e)^2 + 4b > 0, the roots are real.But for the system to be stable, we need the real parts of the eigenvalues to be negative.If the roots are real, then both roots must be negative. So, for real roots, we need:1. Sum of roots = (a + e) < 02. Product of roots = -(b - ea) > 0 => b - ea < 0 => ea > bIf the roots are complex, then the real part is (a + e)/2, which must be negative. So, (a + e) < 0.Additionally, for complex roots, the discriminant must be negative, but in our case, the discriminant is (a - e)^2 + 4b. So, if (a - e)^2 + 4b < 0, which would require (a - e)^2 < -4b. But since (a - e)^2 is non-negative, this would require b < 0 and |b| > (a - e)^2 /4.But in the problem statement, it's given that the system exhibits stable dynamics, so we can assume that the eigenvalues have negative real parts, regardless of whether they are real or complex.Therefore, the conditions for the system to reach a steady state are:1. The determinant Δ = a e - b d ≠ 0, so that a unique steady state exists.2. The real parts of the eigenvalues are negative, which requires: a. If the eigenvalues are real: (a + e) < 0 and ea > b b. If the eigenvalues are complex: (a + e) < 0But since the system is stable, we can just state that the real parts of the eigenvalues are negative, which is guaranteed if (a + e) < 0 and the discriminant is such that the roots are either both negative (if real) or have negative real parts (if complex).But perhaps it's more straightforward to state the conditions as:For the system to reach a steady state, the determinant a e - b d ≠ 0, and the real parts of the eigenvalues of the homogeneous system are negative, which requires that (a + e) < 0 and (ea - b) > 0.Wait, because the product of the roots is (ea - b). For real roots, the product must be positive for both roots to be negative (since sum is negative). For complex roots, the product is still (ea - b), but since they are complex conjugates, their product is positive, so (ea - b) > 0 is still required.Therefore, the conditions are:1. a e ≠ b d (for a unique steady state)2. a + e < 0 (sum of eigenvalues negative)3. ea - b > 0 (product of eigenvalues positive)So, these are the conditions on the constants.To summarize, the steady state exists and is stable if:- a e ≠ b d- a + e < 0- ea > bSo, that's the answer for the second sub-problem.Final Answer1. The general solution is:[N_1(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} + frac{ec + bf}{b - ea}][N_2(t) = left( frac{a - r_1}{b} C_1 right) e^{r_1 t} + left( frac{a - r_2}{b} C_2 right) e^{r_2 t} + frac{a(ec + bf) + bc}{b(b - ea)}]where ( r_1 ) and ( r_2 ) are the roots of the characteristic equation ( r^2 - (a + e)r - (b - ea) = 0 ), and ( C_1 ) and ( C_2 ) are constants determined by initial conditions.2. The conditions for the nutrients to reach equilibrium are:[boxed{a + e < 0}, quad boxed{ea > b}, quad text{and} quad boxed{ae neq bd}]</think>

🧠:<think>Okay, so I have this quantum mechanics problem here, and I need to figure it out step by step. Let me start by reading the problem again to make sure I understand what's being asked.The wave function given is Ψ(x, t) = A e^(-α x²) e^(-i β t). I need to normalize this wave function over all space at time t = 0, which means I have to find the normalization constant A. Then, given that α and β are related by α = (m β)/ℏ, I have to calculate the expectation value of the position operator x̂ for the normalized wave function at any time t.Alright, let's tackle the first part: normalizing the wave function.Normalization means that the integral of the absolute square of the wave function over all space should equal 1. So, the integral from negative infinity to positive infinity of |Ψ(x, t)|² dx should be 1.Since the wave function is Ψ(x, t) = A e^(-α x²) e^(-i β t), the absolute square is |Ψ|² = A² e^(-2 α x²), because the exponential with the imaginary unit will have a magnitude of 1 when squared.So, the integral becomes:∫_{-∞}^{∞} A² e^(-2 α x²) dx = 1I remember that the integral of e^(-a x²) dx from -∞ to ∞ is √(π/a). So, in this case, a is 2α, so the integral becomes √(π/(2α)).Therefore, A² * √(π/(2α)) = 1So, solving for A:A² = 1 / √(π/(2α)) = √(2α/π)Therefore, A = (2α/π)^(1/4)Wait, hold on, let me double-check that. If A² * √(π/(2α)) = 1, then A² = 1 / √(π/(2α)) = √(2α/π). So, A is the square root of that, which is (2α/π)^(1/4). Yeah, that seems right.Alternatively, sometimes people write it as A = (α/π)^(1/4) * √√2, but either way, it's the same.So, that's part one done. Now, moving on to part two: calculating the expectation value of the position operator x̂.The expectation value of x̂ is given by the integral of Ψ*(x, t) x Ψ(x, t) dx over all space.But wait, the wave function is Ψ(x, t) = A e^(-α x²) e^(-i β t). So, Ψ*(x, t) is A e^(-α x²) e^(i β t).Multiplying Ψ* and Ψ, we get A² e^(-2 α x²). Then, multiplying by x, the integrand becomes A² x e^(-2 α x²).So, the expectation value ⟨x⟩ is:∫_{-∞}^{∞} A² x e^(-2 α x²) dxHmm, but wait, the integrand is an odd function because x is odd and e^(-2 α x²) is even. So, the product is odd. The integral of an odd function over symmetric limits is zero.Therefore, ⟨x⟩ = 0.But let me think again. Is that correct? Because the wave function is symmetric around x=0, so the expectation value of position should indeed be zero. That makes sense because the wave function is even in x, so there's no preferred direction.But just to make sure, let me write it out:⟨x⟩ = ∫_{-∞}^{∞} Ψ*(x, t) x Ψ(x, t) dx= ∫_{-∞}^{∞} A² x e^(-2 α x²) dxSince the integrand is odd, the integral is zero.So, the expectation value is zero.But wait, the problem says that α and β are related by α = (m β)/ℏ. Does that affect the expectation value? Hmm, in this case, no, because the expectation value of x is zero regardless of the parameters, as long as the wave function is symmetric.But maybe I should check if the wave function is indeed symmetric. The wave function is A e^(-α x²) e^(-i β t). The spatial part is e^(-α x²), which is even in x, so yes, symmetric. So, the expectation value of x is zero.Therefore, the expectation value ⟨x⟩ is zero.Wait, but just to make sure, let me think about the time dependence. The wave function has a time factor e^(-i β t), but when taking the expectation value, it's multiplied by its complex conjugate, which gives e^(i β t) e^(-i β t) = 1. So, the time dependence cancels out, and the expectation value is the same at any time t.So, yeah, it's zero.So, summarizing:1. The normalization constant A is (2α/π)^(1/4).2. The expectation value of x is zero.I think that's it. Let me just recap the steps to make sure I didn't skip anything.For normalization:- Compute the integral of |Ψ|² dx from -∞ to ∞.- Substitute |Ψ|² = A² e^(-2 α x²).- Use the Gaussian integral formula ∫ e^(-a x²) dx = √(π/a).- Solve for A, getting A = (2α/π)^(1/4).For expectation value:- Write the expression for ⟨x⟩.- Notice that the integrand is odd, so integral is zero.- Confirm that the wave function is symmetric, so ⟨x⟩ must be zero.Yep, that all checks out.Final Answer1. The normalization constant is (boxed{A = left( frac{2alpha}{pi} right)^{1/4}}).2. The expectation value of the position operator is (boxed{0}).</think>