Find the particular solution of the following differential equation, given that y=1 , when x=0, (1+x^2 ) d y d x =e^ m ( ⁻¹ x ) -y

Step-by-step solution

Given, differential equation (1+x^2 ) d y d x =e^ m ( ⁻¹ x ) -y, y(0)=1 d y d x = e^ m ⁻¹ x 1+x^2 - y 1+x^2 d y d x +y ( 1 1+x^2 )= e^ m ⁻¹ x 1+x^2 ...(i) On comparing with Bernoulli's equation d y d x +P y=Q P= 1 1+x^2 , Q= e^ m ⁻¹ x 1+x^2 I F=e^ P d x =e^ 1 1+x^2 d x =e^ ⁻¹ x Solution y Eq. (i), we get y I F= Q I F d x+C y e^ ⁻¹ x = e^ m ⁻¹ x 1+x^2 e^ ⁻¹ x d x+C Put ⁻¹ x=t aligned & 1 1+x^2 d x=d t & y e^ ⁻¹ x = e^ m t e^t d t+C & y e^ ⁻¹ x = e^ (m+1) t m+1 +C aligned = e^ (m+1) ⁻¹ x m+1 +C y=1 , when, x=01 e^0=