Question

Solve $\left(\frac{dy}{dx}\right)+\frac{y}{x}=x^3$:

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

Answer 1

The correct option is A

$y=\frac{x^4}{5}+\frac{c}{x}$

Explanation for correct option

Solving non-homogeneous first order linear differential equation:

The given differential equation is: $\left(\frac{dy}{dx}\right)+\frac{y}{x}=x^3...\left(i\right)$

Comparing the above equation $\left(i\right)$ to the standard form $\frac{dy}{dx}+py=Q$:

We get, $P=\frac{1}{x},Q=x^3$

Calculating the integrating factor:

$$\begin{gathered} I.F.=e^{\int pdx} \\ =e^{\int \frac{1}{x}dx} \\ =e^{{\log }_e\left(x\right)} \\ =x \end{gathered}$$

We know that the general solution of non-homogenous first-order linear differential equation is:

$P\left(I.F.\right)=\int Q\left(I.F.\right)dx$

Therefore, we can write:

,$\begin{array}{l}yx=\int x^3\left(x\right)dx+C\\ \Rightarrow yx=\int x^{4}dx+C\\ \Rightarrow yx=\frac{x^5}{5}+C\\ \Rightarrow y=\frac{x^4}{5}+\frac{C}{x}\end{array}$

Hence, the correct answer is Option (A).