Question

please help me to find the solution of this question​

Answer 1

Given---> x dy / dx - y = x² Sinx

To find ---> Solution of given differential equation

Solution---> ATQ,

x dy / dx - y = x² Sinx

Dividing whole equation by x we get

=> dy / dx - y / x = x² Sinx / x

=> dy / dx + (- 1 / x ) y = x Sinx ....................( 1 )

Now , it is a linear differetial equation in x .

So comparing it with

dy / dx + P y = Q

P = - ( 1 / x ) , Q = x Sinx

Integrating factor ( I. F. ) = e^ ( ∫ p dx )

= e^ ( ∫ - ( 1 / x ) dx )

= e^ ( - logx )

= e^ ( log x⁻¹ )

We have a property of log as follows

eˡᵒᵍᵖ = p , applying it here we get

= x⁻¹

Solution of linear differetial equation is

y ( I. F. ) = ∫ Q ( I. F. ) dx + c

So solution of given equation is

y ( x⁻¹ ) = ∫ (x Sinx) ( x⁻¹ ) dx + c

= ∫ x⁰ Sinx dx + c

= ∫ 1 ( Sinx ) dx + c

y / x = ∫ Sinx dx + c

y / x = - Cosx + c

y = - x Cosx + cx