Q120 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

The general solution of the differential equation $\frac{d y}{d x}+\frac{y}{x}=x^2$ is

  • A
    $x y=\frac{x^2}{2}+c$
  • B
    $x y=\frac{x^3}{3}+c$
  • Correct answer
    C
    $x y=\frac{x^4}{4}+c$
  • D
    $x y=\frac{x^5}{5}+c$

Solution

$\frac{d y}{d x}+\frac{y}{x}=x^2$ $\begin{aligned} & \Rightarrow \quad \frac{d y}{d x}+\frac{1}{x} y=x^2 \\ & \Rightarrow \quad P=\frac{1}{x}, Q=x^2 \\ & \Rightarrow \quad I F=e^{\int P d x}=e^{\int \frac{1}{x} d x}=e^{\ln x}=x \\ & \Rightarrow \quad y \cdot I F=\int Q \cdot I F d x+C \\ & \Rightarrow \quad x y=\int x^2 \cdot x d x+C \\ & \therefore \quad x y=\frac{x^4}{4}+C .\end{aligned}$

Hence, the correct answer is option: 'C'.

Relevant Questions