The general solution of a differential equation in a form of `y' = f(x) ` can
be evaluated using direct integration. The derivative of y denoted as` y'` can be written as `(dy)/(dx)` then `y'= f(x)` can be expressed as `(dy)/(dx)= f(x)` .
For the problem `yy'=4sin(x)` , we may apply `y' = (dy)/(dx) ` to set-up the integration:
`y(dy)/(dx)= 4sin(x)` .
or `y dy = 4 sin(x) dx`
Then set-up direct integration on both sides:
`inty dy = int 4 sin(x) dx`
Apply Power Rule integration: `int u^n du= u^(n+1)/(n+1) ` on `inty dy` .
Note: `y` is the same as `y^1` .
`int y dy = y^(1+1)/(1+1)`
Apply the basic integration property: ` int c*f(x)dx= c int f(x) dx` and basic integration formula for sine function: `int sin(u) du = -cos(u) +C`
`int 4 sin(x) dx= 4int sin(x) dx`
`= -4 cos(x) +C`
Then combining the results for the general solution of differential equation:
`y^2/2 = -4cos(x)+C`
`2* [y^2/2] = 2*[-4cos(x)]+C`
`y = +-sqrt(C-8cosx)`