Differential Equations
Linear ODE — $\ln x$ integrating factor
nta_pyq_2024_apr
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $(2x\log_e x)\dfrac{dy}{dx}+2y=\dfrac{3}{x}\log_e x$, $x>0$ and $y(e^{-1})=0$. Then $y(e)$ is equal to:
$-\dfrac{3}{e}$
$-\dfrac{3}{2e}$
$-\dfrac{2}{3e}$
$-\dfrac{2}{e}$
Step-by-Step Solution
Key Concept: Rewrite: $\frac{dy}{dx}+\frac{y}{x\ln x}=\frac{3}{2x^2}$. IF $=\ln x$. $y\ln x=\int\frac{3\ln x}{2x^2}dx$.
$y(e)=-3/e$.
Correct Answer: 1