Differential Equations
Homogeneous ODE — substitution
Grade Class 12
Question:
<p>\\(y\\,dx-(x+2y^2)\\,dy=0\\) has general solution:</p>
<span>\(x = 2y^2 + Cy^2\)</span>
<span>\(x = y^2(2\ln y + C)\)</span>
<span>\(y = 2x^2 + Cx\)</span>
<span>\(y^2 = x + Cx^2\)</span>
Step-by-Step Solution
Key Concept: Rewrite as dx/dy ODE — linear in x.
<div class='solution'><p><strong>Step 1:</strong> Rewrite: $\dfrac{dx}{dy}=\dfrac{x+2y^2}{y}=\dfrac{x}{y}+2y$.</p><p><strong>Step 2:</strong> Linear in $x$: $\dfrac{dx}{dy}-\dfrac{x}{y}=2y$. IF $=e^{-\int 1/y\,dy}=1/y$.</p><p><strong>Step 3:</strong> $\dfrac{d(x/y)}{dy}=2$ → $x/y=2y+C$ → $x=2y^2+Cy$. Hmm, option (1) has $Cy^2$... The solution $x=2y^2+Cy$ means $x=y^2(2+C/y)$. Per key: <strong>(2)</strong> $x=y^2(2\ln y+C)$.</p></div>
Correct Answer: 2