Correct use of separation of variables for second order DE? 
Is this the correct use of separation of variables? I multiped both sides by dx^2 then integrated. Please let me know if you find any mistakes.
Thanks for any help.
 A: Rewrite the DE as:
$$\dfrac {dy'}{dx}=\dfrac 1 {(x+1)^2}+é^{2x}+3$$
And separate :
$$\int  {dy'}=\int  \left ( \dfrac 1 {(x+1)^2}+é^{2x}+3  \right)dx$$
$$  {y'}=- \dfrac 1 {x+1}+\dfrac 12é^{2x}+3x  +C$$
And separate again.
A: A simpler way to express this particular problem might be to consider the equation as:
$$
y''=f(x)
$$
which then leads to:
$$
\int\int y'' dx\ dx=\int\int f(x)\ dx\ dx
$$
and therefore:
$$
\int y'+A\ dx=\int F(x)+B\ dx
$$
and finally:
$$
y+Ax+C=\int F(x)\ dx+Bx+D
$$
where $Ax+C$ can be incorporated on the RHS to have:
$$
y=\int \int f(x)\ dx\ dx+\alpha x+\beta
$$

So why does it work well for first order differential equations? This is because the technique of separation depends on applying integration via substitution, namely:
$$
\frac{dy}{dx}=f(x)\cdot g(y)\iff\frac{y'(t)}{x'(t)}=f(x)\cdot g(y)\iff\int\frac1 {g(y)}\frac{dy}{dt}\ dt=\int f(x)\frac{dx}{dt}\ dt
$$
and then the $dt$'s cancel on both sides due to the substitution. And substitution in turn is based on the chain rule:
$$
\frac{d}{dt}\Phi(\gamma(t))=\varphi(\gamma(t))\cdot\gamma'(t)=\varphi(\gamma)\cdot\frac{d\gamma}{dt}
$$
where the corresponding second order chain rule does not lend itself to the same kind of trick.
