Differential Equations
Implicit ODE via exact differential
MMTS_Full_Test_18
Grade 12
Question:
The curve satisfying $2xy(y^2\cos(x^2y)-1) + x^2y'(y^2\cos(x^2y)+1)=0$ and passing through $(0,1)$ is
(A) $\sin^2(x^2y)=x$
(B) $\sin(x^2y)=xy^2$
(C) $\sin(x^2y)=x^2y$
(D) $\sin(x^2y)=\dfrac{x^2}{y}$
Step-by-Step Solution
Key Concept: Rearrange: $y^2\cos(x^2y)[2xy+x^2y'] = 2xy-x^2y'$. Let $z=x^2y$; then $dz=2xy\,dx+x^2\,dy$. The equation separates into $\cos(z)\,dz = d(x^2/y)$.
$\sin(x^2y)=x^2/y$.
Correct Answer: (D) $\sin(x^2y)=\dfrac{x^2}{y}$