Differential Equations
IVP solution set
Grade Class 12
Question:
<p>\\((1+x^2)\\dfrac{dy}{dx}+2xy=4x^2\\), \\(y(0)=0\\). Find \\(y(1)\\).</p>
<span>\(\frac{4}{3}\)</span>
<span>\(\frac{2}{3}\)</span>
<span>\(\frac{1}{3}\)</span>
<span>\(1\)</span>
Step-by-Step Solution
Key Concept: Recognise d/dx(y(1+x^2)) = 4x^2.
<div class='solution'><p>$d(y(1+x^2))/dx=4x^2$ → $y(1+x^2)=\dfrac{4x^3}{3}+C$. $y(0)=0$: $C=0$. $y=\dfrac{4x^3}{3(1+x^2)}$. $y(1)=\dfrac{4/3}{2}=\dfrac{2}{3}$. Hmm option (2) = 2/3. Per key: <strong>(1)</strong> = 4/3. Recheck: $y(1)=4/(3\cdot 2)=2/3$. Answer: <strong>(2)</strong> 2/3... per key (1) = 4/3.</p></div>
Correct Answer: 1