New to Diff EQ, simple problems

If someone could check these I would be greatly appreciate it:

Find the general solution:

sin^2(x)*(dy/dx) = ( 5+y)^2

(dy/dx)= ((5+y)^2)/sin^2(x)

Move the 5+y squared over and integrate:

-1/(y(x)+5) = -cot(x) + C

**y(x) = ((cot(x) - 5)/(-cot(x)) + C**

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I have no idea how to even begin this.. I've read tuts on here for clues but I really don't know..

C*y'-A*sin(x)*y=b*e^(-Bx)

Constants: A,B,C,a,b

It's the last question of the practice problems but apparently there will be a question like it on the test.

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Theres one other problem I decided to redo before I post it that I would like comments on.

Any help is appreciated.

Re: New to Diff EQ, simple problems

Your integration of the ode is fine, but you've gone astray with the very last line.

It makes life easier if, in this case, you let the constant of integration be $\displaystyle -C.$ That allows you to get rid of the negative signs.

$\displaystyle \frac{1}{y+5}=\cot(x)+C.$

Now if you take the reciprocal both sides (**and that's the whole of the RHS including the $\displaystyle C),$** and shunt the $\displaystyle 5$ to the other side of the equation, you have $\displaystyle y=....$

Your second example is of the integrating factor type. Start by dividing throughout by $\displaystyle C$ and then multiply throughout by the integrating factor

$\displaystyle I=e^{\int p(x) dx}$ where $\displaystyle p(x)$ is the coefficient of $\displaystyle y$ in the second term (after the division by $\displaystyle C$ and including the negative sign).

I fear that you will struggle with the integral of the resulting RHS though. Check that you've copied the question correctly.

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Re: New to Diff EQ, simple problems