Below is my work for a particular problem that is mixing me up, since no matter how many times, I can't get my answer to match the book solution.
Given {f}''(x)= x^{-\frac{3}{2}} where f'(4)= 2 and $f(0)= 0$, solve the differential equation.
f'(x)= \int x^{-\frac{3}{2}} \Rightarrow \frac{x^{-\frac{3}{2}+1}}{-\frac{3}{2}+1} \Rightarrow -2x^{-\frac{1}{2}} + C
f'(4)= -2(4)^{-\frac{1}{2}}+ C= 2 \Rightarrow -4+C= 2 \Rightarrow C= 6 Thus, the first differential equation is f'(x)= -2x^{-\frac{1}{2}}+6
$f(x)= \int -2x^{-\frac{1}{2}}+6 \Rightarrow 2(\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}+1}) \Rightarrow -4x^{\frac{1}{2}}+6x+C$
Since $f(0)= 0, C=0$, so the final differential equation should be $f(x)= -4x^{\frac{1}{2}}+6x$, but the book answer has $3x$ in place of my $6x$. Where did I go wrong?