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1) Solve the differential equation

xdy=(5y+x+1)dx

Here is what I did:

x=(5y+x+1)dx/dy

[tex]x=5xy+\frac{x^2}{2}+x[/tex]

[tex]0=5xy+\frac{x^2}{2}[/tex]

[tex]\frac{-x^2}{2}=5xy[/tex]

[tex]y=\frac{-x}{10}[/tex]

2) Solve: [tex]y(x^2-1)dx+x(x^2+1)dy=0[/tex]

Here is what I did:

[tex]x^2ydx-ydx+x^3dy+xdy=0[/tex]

[tex]\frac{xdy-ydx}{x^2}+ydx+xdy=0[/tex]

[tex]d(\frac{y}{x})+d(xy)=0[/tex]

This is where I am stuck at. I need to have something multiply the first differential that will yield something in the form of [tex]\frac{y}{x}[/tex] and also have the same thing multiply the second differential and will yield something in the form of xy.

3) Find a differential equation with the solution [tex]y=c_1sin(2x+c_2)[/tex]

Here is what I did:

[tex]y'=2c_1cos(2x+c_2)[/tex]

and since

[tex]c_1=\frac{y}{sin(2x+c_2)}[/tex]

[tex]y'=\frac{2ycos(2x+c_2)}{sin(2x+c_2}[/tex]

[tex]y'=2ycot(2x+c_2)[/tex]

I'm not sure what to do after this. I know I need to somehow get rid of the [tex]c_2[/tex] constant, but I don't know how to do this.

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