Name: Variable Separable Form
Uploaded: 2019-07-19
Duration: 9 min 26 s
Description: This video talks about the method of integration by separating variables. The method is explained in an easy-to-understand manner in this video. Watch and le...

Question

Solve the following differential equation :

$x (x - 1) \frac{d y}{d x} - (x - 2) y = x^{3} (2 x - 1)$

Accepted Answer

Given differential equation is $x (x - 1) \frac{d y}{d x} - (x - 2) y = x^{3} (2 x - 1)$ .

So, $\frac{d y}{d x} - \frac{x - 2}{x (x - 1)} y = \frac{x^{2} (2 x - 1)}{(x - 1)}$

Which is a linear differential equation in $y$ .

On Comparing it with $\frac{d y}{d x} + P y = Q$ , We get

$P = \frac{- (x - 2)}{x (x - 1)}, Q = \frac{x^{2} (2 x - 1)}{(x - 1)}$

$IF = e^{\int P d x} = e^{\int \frac{2 - x}{x (x - 1)} d x}$

Consider, $\frac{2 - x}{x (x - 1)} = \frac{A}{x} + \frac{B}{x - 1}$

$\Rightarrow 2 - x = A (x - 1) + B x$

Now, $x = 0 \Rightarrow 2 = - A \Rightarrow A = - 2$

and $x = 1 \Rightarrow B = 1$

So, $\frac{2 - x}{x (x - 1)} = \frac{- 2}{x} + \frac{1}{x - 1}$

Integrating on both sides, we get

$\int \frac{2 - x}{x (x - 1)} d x = - 2 \int \frac{d x}{x} + \int \frac{d x}{x - 1}$

$= - 2 \log x + \log (x - 1) = \log \frac{x - 1}{x^{2}}$

So, $IF = e^{\int \frac{2 - x}{x (x - 1)} d x} = e^{\log \frac{x - 1}{x^{2}}}$

$= \frac{x - 1}{x^{2}}$

The integrating factor, $IF = \frac{x - 1}{x^{2}}$ .

The complete solution is

$y \cdot IF = \int Q \cdot IF d x$

$\Rightarrow y \cdot (\frac{x - 1}{x^{2}}) = \int \frac{x^{2} (2 x - 1)}{(x - 1)} (\frac{x - 1}{x^{2}}) d x$

$\Rightarrow y (\frac{x - 1}{x^{2}}) = \int (2 x - 1) d x$

$\Rightarrow y (\frac{x - 1}{x^{2}}) = x^{2} - x + c$

Which is the required solution of given differential equation.

Solve the following differential equation :

$x (x - 1) \frac{d y}{d x} - (x - 2) y = x^{3} (2 x - 1)$

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Solve the following differential equation : x(x-1)dydx-(x-2)y=x3(2x-1)

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Solve the following differential equation :

$x (x - 1) \frac{d y}{d x} - (x - 2) y = x^{3} (2 x - 1)$