Solve sqrt(16-x^2) between -4 and 0

I am having trouble figuring out a method for finding a solution to
${\displaystyle r+\left(\frac{2}{r}\right){\left(r^{\prime }\right)}^2-r{}^{″}=0}$
I have tried substitution of ${\displaystyle w=r^{\prime }}$ to obtain
${\displaystyle r+\left(\frac{2}{r}\right)w^2=\frac{dw}{dr}w}$
But then I am not sure how to proceed from there.
Any suggestions are welcome. Tha

Find the general solution of the given differential equations
${\displaystyle 4y{}^{″}-4y^{\prime }-3=0}$

Which of the following is not a solution of ${\displaystyle y{}^{″}+4y=0}$
(A) ${\displaystyle 4\cos \left(2x\right)}$
(B) ${\displaystyle 5\sin \left(2x\right)}$
(C) ${\displaystyle \sin \left(2x\right)\cos \left(2x\right)}$
(D) ${\displaystyle 4\cos \left(2x\right)+5\sin \left(2x\right)}$

Find the solution of the following initial value problem:
${\displaystyle x{}^{″}\left(t\right)-7x^{\prime }\left(t\right)+10x\left(t\right)=0,x\left(0\right)=2,x^{\prime }\left(0\right)=3}$

Solve the given differential equation by undetermined coefficients.
${\displaystyle y{}^{″}+4y=4\sin \left(2x\right)}$

Give the correct answer and solve the given equation $xy\prime \prime -y\prime =(3+x)x^2{e}^x$

Cosider the system of differential equations
$x_1^{\prime }=2x_1+x_2$
$x_2^{\prime }=4x_1-x_2$
Convert this system to a second order differential equations and solve this second order differential equations