Solve the following differential equations: x dy dx =x + y

Question

Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$

✓

Answer

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x + vx}}{\text{x}}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=1+\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1+\text{v}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1$
$\Rightarrow\ \text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \text{v}=\log|\text{x}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \frac{\text{y}}{\text{x}}=\log|\text{x}|+\text{C}$
$\Rightarrow\ \text{y}=\text{x}\log|\text{x}|+\text{Cx}$
Hence, $\text{y}=\text{x}\log|\text{x}|+\text{Cx}$ is the required solution.

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