Find the minimum value taken by the following two integral expressions where $\displaystyle y$ and $\displaystyle y'$ are both functions of $\displaystyle x$

(a) $\displaystyle \int_{0}^{1}((y')^2-6y^2)e^{-5x}dx$$\displaystyle , y(0)=1, y'(0)=2$

(b) $\displaystyle \int_{0}^{1}\frac{(y')^2}{(1+y)^2}dx, y(0)=0, y(1)=e^3-1$

I've had a few stabs at this but my answers are looking dodgy. Could someone please show me the way? ^^

Here is what I've done so far:

(a) Using Euler Lagrange Equation $\displaystyle \frac{d}{dx}(\frac{\partial F}{\partial y'})=\frac{\partial F}{\partial y}$, I get

$\displaystyle \frac{d}{dx}(2e^{-5x}y')=-12e^{-5x}y$

$\displaystyle (2e^{-5x}y')=\frac{-12e^{-5x}y}{5}+A$

$\displaystyle y'=\frac{6y}{5}+\frac{A}{2e^{-5x}}$

$\displaystyle y(0)=1, y'(0)=2$ so $\displaystyle A=\frac{8}{5}$

$\displaystyle y'=\frac{dy}{dx}$ so by integrating the last expression for $\displaystyle y'$ I get $\displaystyle y=\frac{6xy}{5}+\frac{4e^{5x}}{25}+B$

Applying boundary conditions again, I get $\displaystyle B=\frac{21}{25}$

Substituting and rearranging, I end up with $\displaystyle y(1-\frac{6x}{5})=\frac{4e^5x+21}{25}\rightarrow y(x)=\frac{4e^5x+21}{25-30x}$ as the extremal funcation.

For part (b) I did something similar using $\displaystyle F-y'\frac{\partial F}{\partial y'}=constant$ as the Euler Langrage equation and ended up with $\displaystyle y=\frac{x}{x-1}$

Here is my working out for part (b):

$\displaystyle I=\int_{0}^{1}\frac{(y')^2}{(1+y)^2}dx, y(0)=0, y(1)=e^3-1$

so using $\displaystyle F-y'\frac{\partial F}{\partial y'}=constant$, I get

$\displaystyle \frac{(y')^2-2(y')^2}{(1+y)^2}=k$

Taking square roots of both sides and multiplying the whole equation by -1, I get

$\displaystyle \frac{y'}{1+y}=A\rightarrow y'=A+Ay \rightarrow y=Ax+Axy+B$

Using boundary condition$\displaystyle y(0)=0$, I get $\displaystyle B=0$

Substituting and rearranging, I get

$\displaystyle y=\frac{Ax}{1+Ax}$

Using the second boundary condition, I get

$\displaystyle e^3-1=\frac{A}{1+A}$

$\displaystyle (e^3-1)(1+A)=A$

$\displaystyle A=\frac{e^3-2}{2-e^3}=-1$

Hence,$\displaystyle y=\frac{x}{x-1}$