Differential Equations
Homogeneous ODE; variable separation
MJMT_Full_Test_10
Grade 12
Question:
A curve passes through $\left(1,\dfrac{\pi}{6}\right)$. Let the slope at each point $(x,y)$ be $\dfrac{y}{x}+\sec\!\left(\dfrac{y}{x}\right)$, $x>0$. The equation of the curve is
$\sin\!\left(\dfrac{y}{x}\right)=\log x+\dfrac{1}{2}$
$\cosec\!\left(\dfrac{y}{x}\right)=\log x+2$
$\sec\!\left(\dfrac{2y}{x}\right)=\log x+2$
$\cos\!\left(\dfrac{2y}{x}\right)=\log x+\dfrac{1}{2}$
Step-by-Step Solution
Key Concept: Let $v=y/x$. ODE: $v+xv'=v+\sec v$ → $\cos v\,dv=dx/x$ → $\sin v=\ln x+C$.
$\sin(y/x)=\ln x+\frac{1}{2}$.
Correct Answer: 1