Find the general solution?

Question

Find the general solution of y'+(2x/(1+x^2))y=e^(-x)/(1+x^2).

 

Answer: y=-(e^(-x)+C)/(1+x^2)

 

Please show all your work step by step.

Answer 1

Hi Sun,

 

The differential equation is in it's standard form:

 

y'+p(x)y=q(x)

 

The general solution of a first order linear differential equation is given by:

 

y = (∫u(x)q(x) dx + C)/u(x)

 

where u(x) is the integrating factor:

 

u(x) = e^{∫p(x) dx}

 

For this problem, p(x) = 2x/(1+x²) and q(x) = e^-x/(1+x²)

 

So, our integrating factor is

 

u(x) = e^{∫2x/(1+x^2) dx}

  

        = e^{ln(1+x^2)}

 

        = 1+x²

 

Plugging this into our general solution yields

 

y = ( ∫(1+x²)*e^-x/(1+x²) dx + C )/(1+x²)

 

   = ( ∫e^-x dx +C )/(1+x²)

 

   = (-e^-x+C)/(1+x²)