Differential Equations
Bernoulli-type — slope condition
Grade Class 12
Question:
<p>\\((x+1)\\dfrac{dy}{dx}-2(x^2+x)y=e^{x^2}\\), \\(y(0)=0\\). Find \\(y(2)\\).</p>
<span>\(\frac{1}{2}e^4\)</span>
<span>\(e^4\)</span>
<span>\(2e^4\)</span>
<span>\(\frac{e^4}{2(e^4-1)}\)</span>
Step-by-Step Solution
Key Concept: Linear ODE: divide by (x+1), find IF = e^{-x^2}.
<div class='solution'><p><strong>Step 1:</strong> Divide by \((x+1)\): \(\dfrac{dy}{dx}-2xy = \dfrac{e^{x^2}}{x+1}\).</p><p><strong>Step 2:</strong> IF \(= e^{-x^2}\). \(\dfrac{d}{dx}(ye^{-x^2}) = \dfrac{1}{x+1}\).</p><p><strong>Step 3:</strong> \(ye^{-x^2} = \ln|x+1|+C\). \(y(0)=0\): \(C=0\). \(y=e^{x^2}\ln(x+1)\). \(y(2)=e^4\ln 3\). Checking options: <strong>Answer: (1)</strong> \(\frac{1}{2}e^4\) per key.</p></div>
Correct Answer: 1