Differential Equations
Linear DE — Integrating Factor
nta_pyq_2024_jan
Grade 12
Question:
If $x=x(t)$ is the solution of the differential equation $(t+1)dx=(2x+(t+1)^4)\,dt$, $x(0)=2$, then $x(1)$ equals
Step-by-Step Solution
Key Concept: Rewrite as $\frac{dx}{dt}-\frac{2x}{t+1}=(t+1)^3$. Find the integrating factor $e^{-2\ln(t+1)}=\frac{1}{(t+1)^2}$, integrate, apply IC to find $c$, then evaluate at $t=1$.
$\frac{dx}{dt}-\frac{2x}{t+1}=(t+1)^3$. I.F.$=\frac{1}{(t+1)^2}$. $\frac{x}{(t+1)^2}=\frac{(t+1)^2}{2}+c$. $x(0)=2\Rightarrow c=\frac{3}{2}$. $x(1)=\frac{2^4}{2}+\frac{3}{2}\cdot4=8+6=14$.
Correct Answer: 14