Solve the given differential equation by separation of variables. x(dy/dx) = 4y.

Question:

Solve the given differential equation by separation of variables.

{eq}x(\frac{\mathrm{d}y}{\mathrm{d}x}) = 4y {/eq}

Solving a Differential Equation by Variables Separable:

If a differential equation can be put in the form {eq}N\left ( y \right )\mathrm{d}y=M\left ( x \right )\mathrm{d}x \cdots \left ( * \right ) {/eq}, then we can solve the differential by integrating each side of the equation {eq}\left ( * \right ) {/eq}. This process is known as Variables Separable Method.

Answer and Explanation:

{eq}\begin{align*} x\frac{\mathrm{d} y}{\mathrm{d} x}&=4y \\ \Rightarrow \frac{\mathrm{d}y}{y}&=4\frac{\mathrm{d}x}{x} \\ \Rightarrow \int \frac{\mathrm{d}y}{y}&=4\int\frac{\mathrm{d}x}{x} \\ \Rightarrow \ln y &= 4\ln x + \ln C &\text{[}C\;\text{is a constant of integration ]} \\ &=\ln x^4+\ln C \\ &=\ln Cx^4 \\ \Rightarrow y&=Cx^4 \end{align*} {/eq}