Couple of Questions Regarding Integration and Checking Work

[kotsumu]

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]

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



now integrate both sides...letting

[Martingale]

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...

.

[Martingale]

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))







for the RHS let


...