Differential Equations
Linear ODE via Substitution — Finding x at Given y
nta_pyq_2023_apr
Grade 12
Question:
Let $x=x(y)$ be the solution of $2(y+2)\ln_e(y+2)\,dx+(x+4-2\ln_e(y+2))\,dy=0$, $y>-1$ with $x(e^4-2)=1$. Then $x(e^9-2)$ is equal to
Step-by-Step Solution
Key Concept: Substitute $u=x+4$, $v=y+2$. The ODE becomes linear in $u(v)$: $\frac{du}{dv}+\frac{1}{2v\ln v}u=\frac{1}{v}$. IF $=(\ln v)^{1/2}$.
$u(y+2)=x+4$. With substitution: $u\cdot(\ln v)^{1/2}=\frac{2}{3}(\ln v)^{3/2}+\frac{14}{3}$. At $y=e^9-2$: $3u=18+\frac{14}{3}\Rightarrow u=\frac{68}{9}$. $x=\frac{32}{9}$.
Correct Answer: 3