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Differential Equations

Homogeneous ODE

MMTS_Full_Test_18

Grade 12

Question:

The solution of $\dfrac{dy}{dx}=\dfrac{y}{x}+\sin\dfrac{y}{x}$ is

$\tan(y/2x)=cx$

$\cot(y/2x)=cx$

$\tan(y/x)=cx$

$\cot(y/x)=cx$

Step-by-Step Solution

Key Concept: Let $v=y/x$; $y=vx$; $dy/dx=v+xdv/dx$

$\frac{dv}{\sin v}=\frac{dx}{x}$. $\int\csc v\,dv=\ln|x|+C$. $\ln|\tan(v/2)|=\ln|x|+C$. $\tan(y/2x)=cx$.

Correct Answer: 3