Differential Equations
Integral Equation — Differentiate and Solve Linear ODE
nta_pyq_2026_jan
Grade 12
Question:
Let $f:[1,\infty)\to\mathbb{R}$ be a differentiable function. If $6\displaystyle\int_1^x f(t)\,dt=3x f(x)+x^3-4$ for all $x\geq1$, then the value of $f(2)-f(3)$ is
Step-by-Step Solution
Key Concept: Differentiate both sides: $6f(x)=3f(x)+3xf'(x)+3x^2\Rightarrow f(x)-xf'(x)=x^2$. Recognize $-\dfrac{d}{dx}\!\left(\tfrac{f}{x}\right)=1$.
$f(x)=-x^2+2x$. $f(2)-f(3)=0-(-3)=3$.
Correct Answer: 2