initial value,Evaluate the integral problems

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February 4th 2007, 02:45 PM
killasnake

initial value,Evaluate the integral problems

Hi I need help with these problems, I just started calc 2 and I have no idea how to do these. :confused:

Solve the initial value problem:
http://img.photobucket.com/albums/v629/killasnake/1.gif

and y=1 when x=0

y(x)= ___________

and

Evaluate the integral:
http://img.photobucket.com/albums/v6...241696img1.gif

Answer: ______ + c

and

Perform the following integration:

http://img.photobucket.com/albums/v629/killasnake/0.gif

Answer: ______ + c
February 4th 2007, 03:52 PM
ThePerfectHacker

You have,
$y'+3y=e^{2x}$
An integrating factor is,
$y=\exp \left( \int 3 dx \right)=e^{3x}$ .
Thus, the general solution is,
$y=e^{-3x}\int e^{3x}e^{2x} dx$
$y=e^{-3x}\int e^{5x} dx$
$y=e^{-3x} \left( \frac{1}{5}e^{5x}+C \right)$
$y=\frac{1}{5}e^{2x}+Ce^{-3x}$
We also know that, $y(0)=1$ .
Thus,
$1=\frac{1}{5}(1)+C(1)$
$C=\frac{4}{5}$ .
Thus,
$\boxed{ y= \frac{1}{5}e^{2x}+\frac{4}{5}e^{-3x} }$
February 4th 2007, 03:54 PM
ThePerfectHacker

You have,
$\int \frac{\cosh \sqrt{x}}{\sqrt{x}}dx=2\int \frac{\cosh \sqrt{x}}{2\sqrt{x}} dx$
Use substitution $u=\sqrt{x}$ .
Thus,
$2\int \cosh u du=2\sinh u+C=2\sinh \sqrt{x}+C$
February 4th 2007, 03:58 PM
ThePerfectHacker

First in,
$\int \frac{\sin (4t-1)}{1-\sin^2 (4t-1)} dt$
Use the linear substitution $x=4t-1$ to get,
$\frac{1}{4} \int \frac{\sin x}{1-\sin^2 x} dx$
Pythagorean identity,
$\frac{1}{4} \int \frac{\sin x}{\cos^2 x} dx$
Let $u=\cos x$ then $u'=-\sin x$ .
Thus,
$-\frac{1}{4} \int \frac{1}{u^2} du$
Thus,
$\frac{1}{4u}+C=\frac{1}{4\cos x}+C=\frac{1}{4}\sec x+C$
Substitute again,
$\frac{1}{4}\sec (4t-1)+C$

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