Differential Equations
Variable Separable with Substitution — Power of a
nta_pyq_2023_apr
Grade 12
Question:
If the solution curve $f(x,y)=0$ of the differential equation $(1+\ln x)\dfrac{dy}{dx}-x\ln x=e^y$, $x>0$, passes through $(1,0)$ and $(a,2)$, then $a^a$ is equal to
$e^{2e^2}$
$e^{e^2}$
$e^{\sqrt{2}e^2}$
$e^{2e^{\sqrt{2}}}$
Step-by-Step Solution
Key Concept: Substitute $t=x\ln x$, so $(1+\ln x)dx=dt$. The equation becomes $\frac{dt}{dy}-t=e^y$ — a linear ODE in $t(y)$.
$x\ln x=ye^y$. At $y=2$: $a\ln a=2e^2\Rightarrow a^a=e^{2e^2}$.
Correct Answer: 1