Differential Equations
Differential Equations
nta_pyq_2025_apr
Grade 12
Question:
Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\displaystyle\int_0^x(t+2)f(t)\,dt$, $x\geq 0$. Then $f(2)$ is equal to ____.
Step-by-Step Solution
Key Concept: Differentiate both sides with respect to $x$ to eliminate the integral; the resulting equation simplifies to $\tfrac{1}{3}\tfrac{dy}{y+1} = \tfrac{dx}{x+2}$, which is separable.
Differentiating both sides:
$4(x+2)f(x)+2(x+2)^2 f'(x)-6(x+2) = 10(x+2)f(x)$.
Dividing by $(x+2)$: $2(x+2)f'(x) = 6(x+2)f(x)-4f(x)+6 \Rightarrow \dfrac{f'(x)}{f(x)+1} = \dfrac{3}{x+2}$.
(After rearrangement.) Integrating: $\ln|f+1| = 3\ln|x+2|+C \Rightarrow f+1 = A(x+2)^3$.
At $x=0$: from original equation $2(4)f(0)=\tfrac{3}{2}\cdot 4 \Rightarrow f(0)=\tfrac{3}{2}$, so $\tfrac{5}{2}=8A \Rightarrow A=\tfrac{5}{16}$.
$f(x) = \dfrac{5}{16}(x+2)^3-1$.
$$f(2) = \frac{5}{16}\cdot 64-1 = 20-1 = 19.$$
Correct Answer: 19