inhomogeneous linear differential equation

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February 1st 2009, 09:38 AM
rajr

inhomogeneous linear differential equation

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.
February 1st 2009, 10:22 AM
Chris L T521

Quote:

Originally Posted by rajr

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: $\frac{\,dy}{\,dx}-\frac{x}{1+x^2}y=\sqrt{\frac{1+x^2}{1-x^2}}$

The integrating factor would be $\varrho\!\left(x\right)=e^{-\int\frac{x}{1+x^2}\,dx}$

By u-substitution, it can be shown that $\varrho\!\left(x\right)=\frac{1}{\sqrt{1+x^2}}$

Thus, the DE becomes $\frac{\,d}{\,dx}\left[\frac{1}{\sqrt{1+x^2}}\cdot y\right]=\frac{1}{\sqrt{1-x^2}}$

It should be pretty straightforward from here.

Does this make sense?
February 1st 2009, 10:32 AM
rajr

hi thx for ur help

should i then integrate both sides to get Y
February 1st 2009, 10:37 AM
anon18

no you just integrate the left side and you should be left with an answer like y=(1+x^2)^-1/2[sin^-1(x)+C]
February 1st 2009, 10:50 AM
rajr

thx for ur help
February 1st 2009, 10:54 AM
Chris L T521

Quote:

Originally Posted by anon18

no you just integrate the left side and you should be left with an answer like y=(1+x^2)^-1/2[sin^-1(x)+C]

No. You integrate both sides to get $\frac{1}{\sqrt{1+x^2}}\cdot y=\sin^{-1}\!\left(x\right)+C\implies y=\sqrt{1+x^2}\left[\sin^{-1}\!\left(x\right)+C\right]$ $\implies y=\sqrt{1+x^2}\sin^{-1}\!\left(x\right)+C\sqrt{1+x^2}$ .

When writing the solution, though, we usually prefer to write it as $y=\sqrt{1+x^2}\left[\sin^{-1}\!\left(x\right)+C\right]$
February 1st 2009, 11:10 AM
rajr

thx for ur help Mr Chris....