Differential Equations
Homogeneous-type ODE; exact differential substitution
MJMT_Full_Test_09
Grade 12
Question:
The solution of $x^2\,dy - y^2\,dx + xy(x-y)\,dy = 0$ is $\ln\left|\dfrac{x-y}{xy}\right| = \dfrac{y^k}{2} + c$, then the value of $k$ is
Step-by-Step Solution
Key Concept: Divide throughout by $x^2y^2$ and recognize the exact differentials $d(1/y)$ and $d(1/x)$ to convert to a separable form.
$d(1/x-1/y)+y(1/y-1/x)dy=0$. Let $u=1/x-1/y$: $du/u=-ydy$... integrating: $\ln|\frac{x-y}{xy}|=\frac{y^2}{2}+c$. So $k=2$.
Correct Answer: 2