b)Determine the general solution of the inhomogeneous linear differential equation
y′ = [xy / (1+x^2)] + SQRT [ (1+x^2) / (1-x^2)]
by the method of integrating factor.
b)Determine the general solution of the inhomogeneous linear differential equation
y′ = [xy / (1+x^2)] + SQRT [ (1+x^2) / (1-x^2)]
by the method of integrating factor.
First, rewrite the DE as follows: $\displaystyle \frac{\,dy}{\,dx}-\frac{x}{1+x^2}y=\sqrt{\frac{1+x^2}{1-x^2}}$
The integrating factor would be $\displaystyle \varrho\!\left(x\right)=e^{-\int\frac{x}{1+x^2}\,dx}$
By u-substitution, it can be shown that $\displaystyle \varrho\!\left(x\right)=\frac{1}{\sqrt{1+x^2}}$
Thus, the DE becomes $\displaystyle \frac{\,d}{\,dx}\left[\frac{1}{\sqrt{1+x^2}}\cdot y\right]=\frac{1}{\sqrt{1-x^2}}$
It should be pretty straightforward from here.
Does this make sense?
No. You integrate both sides to get $\displaystyle \frac{1}{\sqrt{1+x^2}}\cdot y=\sin^{-1}\!\left(x\right)+C\implies y=\sqrt{1+x^2}\left[\sin^{-1}\!\left(x\right)+C\right]$ $\displaystyle \implies y=\sqrt{1+x^2}\sin^{-1}\!\left(x\right)+C\sqrt{1+x^2}$.
When writing the solution, though, we usually prefer to write it as $\displaystyle y=\sqrt{1+x^2}\left[\sin^{-1}\!\left(x\right)+C\right]$