Question

The general solution of the differential equation $\frac {dy}{dx}+\frac {y}{x}=3x$ is

• A. $y=x+ \frac {c}{x}$
• B. $y=x^2 + \frac {c}{x}$
• C. $y=x- \frac {c}{x}$
• D. $y=x^2 - \frac {c}{x}$

Answer 1

The Correct Option is D

Given differential equation is $\frac{d y}{d x}+\frac{y}{x}=3 x$ It is a linear differential equation of the form $\frac{d Y}{d x}+P y=Q$ $\therefore P=\frac{1}{x} \text { and } Q=3 x$ $\therefore IF =e^{\int P d x}=e^{\int \frac{1}{x} d x}$ $=e^{\log x}=x$ $\therefore$ Complete solution is $y x=\int 3 x \times x d x +C$...(i) $\Rightarrow y x=3\left[\frac{x^{3}}{3}\right]+C$ $\Rightarrow y=x^{2}+\frac{C}{x}$ Also, E (i) can be written as $y x=\int 3 x \times x dx-C$ $\Rightarrow y x=x^{3}-C$ $\Rightarrow y= x^{2}-\frac{C}{x}$