Ordinary Differential Equations

**Anteeks** · November 16th 2010, 01:37 PM

Need help with following three differential equations. Help is really appreciated!

First:
Solve the differential equation $y' - 2 y = 3$ , y(0) = 1.

Second:

$\displaystyle \frac{dy}{dx} + 8 \sin(x) y = (x+3)^4 e^{8 \cos(x)}$ , y(0)=5.

Third:

$\displaystyle \frac{dy}{dx} = x^5 (3^2+y^2)$ that have the value 4 when x=0

**mr fantastic** · November 16th 2010, 03:13 PM

Originally Posted by Anteeks

Need help with following three differential equations. Help is really appreciated!

First:
Solve the differential equation $y' - 2 y = 3$ , y(0) = 1.

Second:

$\displaystyle \frac{dy}{dx} + 8 \sin(x) y = (x+3)^4 e^{8 \cos(x)}$ , y(0)=5.

Third:

$\displaystyle \frac{dy}{dx} = x^5 (3^2+y^2)$ that have the value 4 when x=0

The first and third are seperable. The second uses the integrating factor.

If you need more help, please show all your work and say where you're stuck.

**Anteeks** · November 16th 2010, 03:47 PM

Thank you!

First equation:

$e^int{(-2x)dx = e^{(-2x)$

$\displaystyle \frac{dy}{dx}*e^{(-2x))$ $-2y*e^{(-2x)$ $= 3*e^{(-2x)$
-->
$D(y*e^{(-2x))$ $=3*e^{(-2x)$
-->
$y*e^{(-2x)$ = $int(3*e^{(-2x))dy$ $=-6*e^{(-2x)$ $+ C$

$y=-6+C*e^{(2x)$
$y(0)=6$ $--> -6+C*e^0=6, C=11$

Which gives $y=-6+11*e^{(2x)$

Is this right?

Second:
integrating factor = $e^{8 \cos(x)}$

$\displaystyle \frac{dy}{dx} *e^{8 \cos(x$ ) $+ 8 \sin(x) *$ $e^{8 \cos(x)$ $y = (x+3)^4$ $e^{8 \cos(x)}*$ $e^{8 \cos(x)$

Thats about how far I can go, and don't know if its right.

Third:
$\displaystyle \frac{dy}{dx} = x^5 (3^2+y^2)$

Divide with $(3^2+y^2)$ on both rows and integrate both:

$\displaystyle \frac{dy}{dx}/(3^2+y^2) = x^5$

Thats how far I can go.

**HallsofIvy** · November 17th 2010, 04:41 AM

Originally Posted by Anteeks

Thank you!

First equation:

$e^int{(-2x)}dx = e^{(-2x)$

Yes, that first equation is linear and so you can get an integrating factor this way. It is $\int -2 dx= -2x$ , not $\int -2x dx$ . After that you are good

$\displaystyle \frac{dy}{dx}*e^{(-2x))$ $-2y*e^{(-2x)$ $= 3*e^{(-2x)$
-->
$D(y*e^{(-2x))$ $=3*e^{(-2x)$
-->
$y*e^{(-2x)$ = $int(3*e^{(-2x))dy$ $=-6*e^{(-2x)$ $+ C$

until here. $3\int e^{-2x}dx= -\frac{3}{2}e^{-2x}+ C$ , not $-6 e^{-2x}$ .

$y=-6+C*e^{(2x)$
$y(0)=6$ $--> -6+C*e^0=6, C=11$

But mr fantastic told you the equation was separable. It is much easier to do $\frac{dy}{dx}= 2y+ 3$ so that $\frac{dy}{2y+ 3}= \frac{dx}$ . Integrating, $ln(2y+ 3)= x+ C$ so that $2y+ 3= ce^{x}$ .

Which gives $y=-6+11*e^{(2x)$
Is this right?

Well, it is not hard to check yourself: if $y= -6+ 11e^{2x}$ then $y'= -22e^{-2x}$ while $-2y= 12- 22e^{-2x}$ . So $y'- 2y= -22e^{-2x}+ 12- 22e^{-2x}= 12- 44e^{-2x}$ . No, that is not "3" and so your solution does not check.

Second:
integrating factor = $e^{8 \cos(x)}$

No, the integral of 8 sin(x) is -8 cos(x).

$\displaystyle \frac{dy}{dx} *e^{8 \cos(x$ ) $+ 8 \sin(x) *$ $e^{8 \cos(x)$ $y = (x+3)^4$ $e^{8 \cos(x)}*$ $e^{8 \cos(x)$

Thats about how far I can go, and don't know if its right.

Third:
$\displaystyle \frac{dy}{dx} = x^5 (3^2+y^2)$

Divide with $(3^2+y^2)$ on both rows and integrate both:

$\displaystyle \frac{dy}{dx}/(3^2+y^2) = x^5$

Thats how far I can go.

One of the things you should have learned long ago is that $\int \frac{du}{1^2+ u^2}= arctan(u)+ C$ .

$\frac{1}{3^2+ y^2}= \frac{1}{3^2}\frac{1}{1+ \left(\frac{y}{3}\right)^2}$ . Let u= y/3.

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