Differential Equations
Homogeneous / Exact Equation
nta_pyq_2024_apr
Grade 12
Question:
Let $\alpha|x|=|y|e^{xy-\beta}$, $\alpha,\beta\in\mathbb{N}$ be the solution of the differential equation $x\,dy-y\,dx+xy(x\,dy+y\,dx)=0$, $y(1)=2$. Then $\alpha+\beta$ is equal to ________.
Step-by-Step Solution
Key Concept: Rewrite: $\frac{dy}{y}-\frac{dx}{x}+d(xy)=0$. Integrate: $\ln|y|-\ln|x|+xy=C$.
$\alpha=2$, $\beta=2$. $\alpha+\beta=4$.
Correct Answer: 4