Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
Solution of the differential equation $x\cos\left(\frac{x}{y}\right)(ydx + xdy) = y\sin\left(\frac{x}{y}\right)(xdy - ydx)$ is :
$y = c\cos\left(\frac{x}{y}\right)$
$\sec\left(\frac{x}{y}\right) = cxy$
$\frac{y}{x}\sec\left(\frac{y}{x}\right) = c$
none of these
Step-by-Step Solution
Key Concept: Recognize the differential equation structure where ydx + xdy = d(xy) and xdy - ydx = d(xy/x²), then substitute v = x/y to transform into a separable homogeneous equation. The integration of ∫(v sin v - cos v)/(v cos v) dv requires decomposition into ∫dv - ∫(dv/v sin v) to obtain sec(x/y).
Divide both sides by $x^2dx$ to obtain a homogeneous equation. Substitute $y = vx$ so $\frac{dy}{dx} = v + x\frac{dv}{dx}$. This gives $2v\cos v = x\frac{dv}{dx}(v\sin v - \cos v)$. Separating variables and integrating: $\int \frac{2dx}{x} = \int \frac{(v\sin v - \cos v)dv}{v\cos v}$ yields $\sec\frac{y}{x} = cxy$.
Correct Answer: 2