## Couple of Questions Regarding Integration and Checking Work

Limits, differentiation, related rates, integration, trig integrals, etc.
kotsumu
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Joined: Fri Jan 28, 2011 4:55 am
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### Couple of Questions Regarding Integration and Checking Work

Ok, for question 1...
I am given a initial value problem dy/dx = 11sqrt(y)cos(sqrt(y))^2 for x>0 and y(0) = (pi^2)/16
- First I isolated dy and dx respectively ending with...
- (11sqrt(y)cos(sqrt(y))^2)dy = dx
So far so good? Am I approaching this problem right?
- Now, I want to integrate both sides...How exactly do I integrate the y expression? Do I substitute sqrt(y) as u? or cos(sqrt(y)) as u?

For question 2...
I am asked to solve the initial value problem given (xy^2)(dy/dx) = 10+5x and y(1) = 7 for x>0
- I split the formula into (y^2)dy = ((5x+10)/x)dx
- Then I integrated both sides to arrive at the integral of...
- (y^3)/3 = 5(x) + 10ln(|x|) + C
- I plugged in X and got C = 328/3
- Then with the integral equation, I solved for why and plugged in X to get a final solution of
- y = (15x+30ln(|x|)+(328/3))^(1/3)
What am I doing wrong? I plugged 1 and 7 back in 1 in x and did not get 7 back as an answer with my final solution...

For question 3...
I am asked to solve 4sec(x)(dy/dx) = e^(y+sin(x)) given y(0) = -4...
- I am just completely lost for this...

Sorry if it's kinda messy, I tried to make it as organized as possible...

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Couple of Questions Regarding Integration and Checking W

Ok, for question 1...
I am given a initial value problem dy/dx = 11sqrt(y)cos(sqrt(y))^2 for x>0 and y(0) = (pi^2)/16
- First I isolated dy and dx respectively ending with...
- (11sqrt(y)cos(sqrt(y))^2)dy = dx
So far so good? Am I approaching this problem right?
- Now, I want to integrate both sides...How exactly do I integrate the y expression? Do I substitute sqrt(y) as u? or cos(sqrt(y)) as u?
dy/dx = 11sqrt(y)cos(sqrt(y))^2

$\frac{dy}{dx}=11\sqrt{y}\cos^2(\sqrt{y})\Leftrightarrow \frac{dy}{\sqrt{y}\cos^2(\sqrt{y})}=11dx$

now integrate both sides...letting $u=\sqrt{y}$

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Couple of Questions Regarding Integration and Checking W

For question 2...
I am asked to solve the initial value problem given (xy^2)(dy/dx) = 10+5x and y(1) = 7 for x>0
- I split the formula into (y^2)dy = ((5x+10)/x)dx
- Then I integrated both sides to arrive at the integral of...
- (y^3)/3 = 5(x) + 10ln(|x|) + C
- I plugged in X and got C = 328/3
- Then with the integral equation, I solved for why and plugged in X to get a final solution of
- y = (15x+30ln(|x|)+(328/3))^(1/3)
What am I doing wrong? I plugged 1 and 7 back in 1 in x and did not get 7 back as an answer with my final solution...

.
$y = (15x+30\ln(x)+328)^{1/3}$

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Couple of Questions Regarding Integration and Checking W

For question 3...
I am asked to solve 4sec(x)(dy/dx) = e^(y+sin(x)) given y(0) = -4...
- I am just completely lost for this...

Sorry if it's kinda messy, I tried to make it as organized as possible...
4sec(x)(dy/dx) = e^(y+sin(x))

$4\sec(x)\frac{dy}{dx} = e^{y+\sin(x)}$

$4\sec(x)\frac{dy}{dx} = e^{y}e^{\sin(x)}$

$4e^{-y}dy = \cos(x)e^{\sin(x)}dx$

for the RHS let $u=\sin(x)$

...