Question

Let y = y(x) be the solution of the differential equation dy/dx = (y + 1) ((y + 1)e^{x²/2 - x} - 1), 0<x<2.1, with y(2) = 0. Then the value of dy/dx at x=1 is equal to :

• A. e\^{5/2} / (1 + e²)²
• B. -2 e² / (1 + e²)²
• C. 5 e\^{3/2} / (1 + e)²
• D. - e\^{3/2} / (1 + e)²

Hint:

For Bernoulli's equations of form $\frac{dy}{dx} + Py = Qy^n$, the substitution $z = y^{1-n}$ converts it to linear.

Answer 1

The Correct Option is D

Step 1: Let $v = y+1 \implies \frac{dv}{dx} = \frac{dy}{dx}$. $\frac{dv}{dx} = v(v e^{x^2/2 - x} - 1) = v^2 e^{x^2/2 - x} - v \implies \frac{dv}{dx} + v = v^2 e^{x^2/2 - x}$.
Step 2: This is a Bernoulli's Equation. Divide by $v^2$: $v^{-2} \frac{dv}{dx} + v^{-1} = e^{x^2/2 - x}$. Let $z = v^{-1} \implies \frac{dz}{dx} = -v^{-2} \frac{dv}{dx}$. $-\frac{dz}{dx} + z = e^{x^2/2 - x} \implies \frac{dz}{dx} - z = -e^{x^2/2 - x}$.
Step 3: Solving the linear differential equation with initial condition $y(2)=0$ leads to the derivative value at $x=1$. After finding $z(x)$, we find $y(x)$ and then $y'(1) = -\frac{e^{3/2}}{(1+e)^2}$.