Question

Solve x (x-1)dy/dx-(x-2)y=x3(2x-1)

Answer 1

The differential equation  x (x-1)dy/dx-(x-2)y = x³(2x-1) can be solved in the following ways:

• As per the question,

x (x-1)dy/dx - (x-2)y = x³(2x-1)

• Dividing both sides by x(x-1), we get,

dy / dx - (x-2) / x(x-1) = x²(2x-1) / (x-1)

• Let,  a = $e^{\int\limits {P} \, dx }$  , where P = -(x-2) / x(x-1)

• Now, Solving ${\int\limits {P} \, dx }$ :

= - ∫ [(x-1) + (-1)] / x(x-11)

= ∫ - 1/ x  + ∫ 1 / (x² - x)

= - ln x + ln (x-1) / x

= ln (x-1)/x²

• Therefore we get   =    $e^{ ln (x-1) / x^{2} }$

                                        =  (x-1)/x²

• Now, y * a = x²(2x-1) / (x-1) * a

         y * (x-1)/x²  =  ∫ x²(2x-1) / (x-1)  * (x-1)/x²

                           = x² - x + C

• Therefore, y = x² / (x-1) * ( x² - x + C )