Question

If $x \sin(\frac{y}{x})dy = (y \sin(\frac{y}{x})-x)dx$ and $y(1) = \frac{\pi}{2}$ then the value of $cos(\frac{y}{e})$ is _______.

Answer 1

1

Answer 2

1. Rewrite the differential equation in the form $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{\sin(\frac{y}{x})}$.
2. Identify it as a homogeneous differential equation and substitute $y=vx$, which implies $\frac{dy}{dx} = v + x\frac{dv}{dx}$.
3. The substitution leads to $x\frac{dv}{dx} = -\frac{1}{\sin(v)}$.
4. Separate variables to get $\sin(v)dv = -\frac{dx}{x}$.
5. Integrate both sides: $\int \sin(v)dv = -\int \frac{dx}{x}$, yielding $-\cos(v) = -\ln|x| + C$.
6. Simplify to $\cos(v) = \ln|x| - C$. Substitute back $v = \frac{y}{x}$ to get $\cos(\frac{y}{x}) = \ln|x| + C_1$.
7. Apply the initial condition $y(1) = \frac{\pi}{2}$: $\cos(\frac{\pi/2}{1}) = \ln|1| + C_1 \implies 0 = 0 + C_1 \implies C_1 = 0$.
8. The particular solution is $\cos(\frac{y}{x}) = \ln|x|$.
9. Evaluate $\cos(\frac{y}{e})$ by substituting $x=e$: $\cos(\frac{y}{e}) = \ln|e| = 1$.