Question

Find the general solution of the Differential Equation $\frac{dy}{dx}-\frac{y}{x}=0.$

Answer 1

Step 1: Find the general solution of $\frac{dy}{dx}-\frac{y}{x}=0.$

Given,

$$\begin{gathered} \Rightarrow \frac{dy}{dx}-\frac{y}{x}=0 \\ \Rightarrow \frac{dy}{dx}=\frac{y}{x} \\ \Rightarrow \frac{dy}{y}=\frac{dx}{x} \end{gathered}$$

Step 2: Using the variable separable method

$$\begin{gathered} \Rightarrow \frac{dy}{y}=\frac{dx}{x} \\ \Rightarrow \int \frac{dy}{y}=\int \frac{dx}{x} \\ \Rightarrow \log \left(y\right)=\log \left(x\right)+\log \left(c\right) \\ \Rightarrow \log \left(y\right)=\log \left(xc\right) \\ \Rightarrow y=xc \end{gathered}$$

Hence, the required general solution is $y=xc$.