Diff equation please

**simfonija** · August 26th 2006, 03:03 PM

(a) 1/x * dy/dx + 2/(1+x^2)*y=3 y(1)=4

(b) 2*x * dy/dx = 2*y + (x^2+y^2)^1/2

Thank you very much

**galactus** · August 26th 2006, 04:09 PM

I'll give you the solution to b, you try to work to that end. Okey-doke?.

b): $\frac{5}{x^{2}+1}+\frac{3(x^{2}+1)}{4}$

**ThePerfectHacker** · August 26th 2006, 04:49 PM

Originally Posted by simfonija

(a) 1/x * dy/dx + 2/(1+x^2)*y=3 y(1)=4

$\frac{1}{x}\frac{dy}{dx}+\frac{2}{1+x^2}y=3$
Multiply by "x",
$\frac{dy}{dx}+\frac{2x}{1+x^2}y=3x$
This is a Linear-First Order Differential Equation.
Here,
$P(x)=\frac{2x}{1+x^2}$
$Q(x)=3x$
Therefore, $u=1+x^2$
$I(x)=\exp \left( \int \frac{2x}{1+x^2} dx\right) =e^{\ln (1+x^2)}=1+x^2$
Thus, the solutions are,
$\frac{1}{I(x)}\cdot \int I(x)Q(x) dx$
Thus,
$(1+x^2)^{-1}\int \frac{3x}{1+x^2} dx$
Use, $u=1+x^2$ ,
$(1+x^2)^{-1}(3/2\ln (1+x^2)+C)$
Now you probably mean,
$y(0)=4$ because these are usually how initial conditions are given.
Thus,
$C=4$
From here the unique solution is,
$\frac{3\ln (1+x^2)}{2 (1+x^2)}+\frac{4}{1+x^2}$

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