Solve the following differential equation (x-1) dy dx =2x^2y - Vidyadip

Question

Solve the following differential equation
$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2\text{y}$

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Answer

We have

$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2\text{y}$

$\Rightarrow\frac{1}{\text{y}}\text{dy}=\frac{2\text{x}^3}{\text{x}-1}\text{dx}$

Integrating both sides, we get

$\int\frac{1}{\text{y}}\text{dy}=\int\frac{2\text{x}^3}{\text{x}-1}\text{dx}$

$\Rightarrow\log|\text{y}|=2\int\frac{\text{x}^3-1+1}{\text{x}-1}\ \text{dx}$

$\Rightarrow\log|\text{y}|=2\int\frac{(\text{x}-1)(\text{x}^2+\text{x}+1)+1}{\text{x}-1}\ \text{dx}$

$\Rightarrow\log|\text{y}|=2\int(\text{x}^2+\text{x}+1)\text{dx}+2\int\frac{1}{\text{x}-1}\ \text{dx}$

$\Rightarrow\log|\text{y}|=\frac{2}{3}\text{x}^3+\text{x}^2+2\text{x}+\log|\text{x}-1|+\text{C}$

Hence, $\log|\text{y}|=\frac{2}{3}\text{x}^3+\text{x}^2+2\text{x}+\log|\text{x}-1|+\text{C}$ is the required solution.

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