Extra problems in calculus of variations January $11,2024$

Solve the problem:

${\displaystyle \underset{1}{\overset{2}{}}}{t}^2{x}^{}{?}^{2}dt$min

$x(1)=2,x(2)=\frac{3}{2}$

Answer: $x=\frac{1}{t}+1in$ absmin$.$

$2.$ Find stationary functions for the problem:

${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}(x^{}{?}^2-x^2+2tx)dt$min

$x(0)=1,x(\frac{}{2})=\frac{}{2}$

Answer: $x=t+cost.$

$3.$ Solve the problem:

${\displaystyle \underset{0}{\overset{1}{}}}(e^{-t}{x}^{}{?}^2-2x)dt$min

$x(0)=0,x(1)=-1$

Answer: $x=(1-t)e^t-1in$ absmin$.$

$4.$ Find stationary functions for the problem:

${\displaystyle \underset{0}{\overset{1}{}}}\frac{x^{}{?}^2}{(1+x{)}^4}dt$min

$x(0)=1,x(1)=0$

Answer: $x=\frac{1-t}{1+t}.$

$5.$ Solve the problem:

${\displaystyle \underset{0}{\overset{1}{}}}{e}^{-x^{}}dt$min

$x(0)=0,x(1)=1$

Answer: $x=$tin absmin$.$