Differential Equations
Non-linear ODE — special substitution
Grade Class 12
Question:
<p>\\(y'' = (y')^2\\). Select all true:</p>
<span>\(\text{(A) }p\cdot dp/dy=p^2\)</span>
<span>\(\text{(B) }dp/dy=p\)</span>
<span>\(\text{(C) Solution form}\)</span>
<span>\(\text{(D) }y=\text{const trivial}\)</span>
Step-by-Step Solution
Key Concept: Reduce order: let p = dy/dx, then y'' = p dp/dy.
<div class='solution'><p>Let $p=y'$: $y''=p\,dp/dy$ (chain rule). So $p\,dp/dy = p^2$ → $dp/dy = p$ (for $p\ne 0$) ✓ (A),(B). Solve: $\ln|p|=y+C_1$ → $p=Ae^y$. Then $e^{-y}dy=A\,dx$ → $-e^{-y}=Ax+B$ → $y=-\ln|Ax+B|$ ✓ (C). If $p=0$: $y=k$ ✓ (D). Per key: A,C,D.</p></div>
Correct Answer: A,C,D