Chapter 7.1, Problem 1E

Use Theorem 7.1 to prove that the boundary value problems have unique solutions $y\left(t\right)$.

1. $\{\begin{array}{c}{y^{\prime }}^{\prime }=y+\frac{2}{3}{e}^t\\ y\left(0\right)=0\\ y\left(1\right)=\frac{1}{3}e\end{array}$
2. $\{\begin{array}{c}{y^{\prime }}^{\prime }=\left(2+4t^2\right)y\\ y\left(0\right)=1\\ y\left(1\right)=e\end{array}$
3. $\{\begin{array}{c}{y^{\prime }}^{\prime }=4y+\left(1-4t\right)e^{-2t}\\ y\left(0\right)=0\\ y\left(1\right)=\frac{1}{2}{e}^{-2}\end{array}$
4. $\{\begin{array}{c}{y^{\prime }}^{\prime }=y+1/t+2{\left(y-e^t\right)}^3\\ y\left(1\right)=e-1\\ y\left(2\right)=e^2-\frac{1}{2}\end{array}$