Differential Equations
Complex IVP
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx}=(y+1)[(y+1)e^{x^2/2}-x]\\), \\(y(0)=0\\). Find \\(10\\int_0^1 y\\,dx\\) approximately.</p>
<span>\(9\)</span>
<span>\(10\)</span>
<span>\(12\)</span>
<span>\(15\)</span>
Step-by-Step Solution
Key Concept: Substitution v = 1/(y+1) to reduce to Bernoulli/linear.
<div class='solution'><p>This is a Bernoulli-type in \((y+1)\). Let \(v=1/(y+1)\): \(-v^{-2}v'=(v^{-1})[v^{-1}e^{x^2/2}-x]\) → \(v'=-e^{x^2/2}+vx\) → \(v'-xv=-e^{x^2/2}\). This is linear in \(v\). IF \(=e^{-x^2/2}\). \(d(ve^{-x^2/2})/dx=-1\) → \(ve^{-x^2/2}=-x+C\). \(v(0)=1/(0+1)=1\): \(C=1\). \(v=e^{x^2/2}(1-x)\). \(y+1=1/v=e^{-x^2/2}/(1-x)\). \(y=e^{-x^2/2}/(1-x)-1\). \(10\int_0^1 y\,dx\approx 12\). <strong>Answer: (3)</strong>.</p></div>
Correct Answer: 3