Differential Equations Question - None

Question

We have, $\frac{dy}{dx} + \left(\frac{-1}{x}\right)y = x\left(xe^x + e^x - 1\right)$. If $y(x=1) = e-1$, find $k$ such that $y(2) = k \cdot y(1) \cdot [y(1)+2]$, and compute $\frac{k^2}{5}$.

Accepted Answer

Step 1: Identify the standard form

The equation is: dy/dx + (-1/x)y = x(xe^x + e^x - 1)

This is a linear ODE with P(x) = -1/x and Q(x) = x(xe^x + e^x - 1) = x²e^x + xe^x - x

Step 2: Find the integrating factor

μ(x) = e^(∫(-1/x)dx) = e^(-ln|x|) = 1/x

Step 3: Multiply both sides by μ(x) = 1/x

(1/x)dy/dx - (1/x²)y = xe^x + e^x - 1

Step 4: Recognize left side as a product derivative

d/dx[y/x] = xe^x + e^x - 1

Step 5: Integrate both sides

y/x = ∫(xe^x + e^x - 1)dx

For ∫xe^x dx, use integration by parts: u = x, dv = e^x dx → du = dx, v = e^x

∫xe^x dx = xe^x - e^x

Therefore: y/x = (xe^x - e^x) + e^x - x + C = xe^x - x + C

So: y = x²e^x - x² + Cx

Step 6: Apply initial condition y(1) = e - 1

e - 1 = 1·e^1 - 1 + C·1

e - 1 = e - 1 + C

C = 0

Step 7: Write the particular solution

y = x²e^x - x²

Step 8: Find y(1)

y(1) = 1·e - 1 = e - 1

Step 9: Find y(2)

y(2) = 4e² - 4

Step 10: Use the relation y(2) = k·y(1)·[y(1) + 2]

4e² - 4 = k(e - 1)[(e - 1) + 2]

4e² - 4 = k(e - 1)(e + 1)

4(e² - 1) = k(e² - 1)

4 = k

Step 11: Calculate k²/5

k²/5 = 16/5 = 3.2

∴ Answer: 3.20

Question:

Step-by-Step Solution

Master Differential Equations with Mathbee