Differential Equations
Limit Condition — Deriving DE via L'Hôpital
nta_pyq_2026_jan
Grade 12
Question:
Let $y=y(x)$ be a differentiable function in the interval $(0,\infty)$ such that $y(1)=2$, and $\displaystyle\lim_{t\to x}\left(\frac{t^2 y(x)-x^2 y(t)}{x-t}\right)=3$ for each $x>0$. Then $2y(2)$ is equal to
Step-by-Step Solution
Key Concept: Apply L'Hôpital w.r.t. $t$: limit $=\dfrac{2ty(x)-x^2y'(t)}{-1}\Big|_{t=x}=x^2y'(x)-2xy(x)=3$. Rearrange: $\dfrac{d}{dx}\!\left(\tfrac{y}{x^2}\right)=\dfrac{3}{x^4}$.
$y(x)=3x^2-\tfrac{1}{x}$. $2y(2)=23$.
Correct Answer: 1