Differential Equations
ODE from Integral Equation — Linear in f
nta_pyq_2023_apr
Grade 12
Question:
Let $f$ be differentiable with $x^2f(x)-x=4\displaystyle\int_0^x tf(t)\,dt$, $f(1)=\dfrac{2}{3}$. Then $18f(3)$ is equal to
Step-by-Step Solution
Key Concept: Differentiate: $x^2f'(x)+2xf(x)-1=4xf(x)\Rightarrow x^2f'(x)-2xf(x)=1$. Linear ODE: $f'-\frac{2}{x}f=\frac{1}{x^2}$. IF $=\frac{1}{x^2}$.
$f(x)=x^2-\frac{1}{3x}$. $18f(3)=18(9-\frac{1}{9})=160$.
Correct Answer: 2