Variational calculus

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Problem: Find admissible extremals of the functional: \[ \begin{array}{l} F(x)=\int_{0}^{\frac{\pi}{2}}\left(\dot{x}^{2}-2 \cos t x\right) d t \rightarrow \text { extr }, \\ x(0)=1, \quad x\left(\frac{\pi}{2}\right)=0 . \end{array} \]

4.1 Variational calculus

2.41 $

Problem: Find admissible extremals of the functional: \[ \begin{array}{l} F(x)=\int_{-1}^{1}\left(\dot{x}^{2}+x^{2}-2 t x\right) d t \rightarrow \text { extr }, \\ x(-1)=-1, \quad x(1)=1 . \end{array} \]

4.2 Variational calculus

2.41 $

Problem: Find admissible extremals in the problem with free endpoints: \[ F(x)=\int_{1}^{2}\left(\dot{x}^{2}+4 x\right) d t \rightarrow \text { extr }, x(1)=1 . \]

4.3 Variational calculus

2.41 $

Problem: Find admissible extremals in the problem with free endpoints: \[ \begin{array}{l} F(x)=\int_{0}^{1}\left(\dot{x}^{2}+4 x^{2}\right) d t \rightarrow \text { extr }, \\ x(1)=\frac{e^{2}+e^{-2}}{2} . \end{array} \]

4.4 Variational calculus

2.41 $

Problem: Find the strong extrema of the functional: \[ \begin{array}{l} F(x)=\int_{0}^{1}\left(\dot{x}^{2}-x^{2}\right) d t \rightarrow \text { extr }, x(0)=1, \\ x(1)=\cos 1 . \end{array} \]

4.5 Variational calculus

3.62 $

Problem: Find the strong extrema of the functional: \[ \begin{array}{l} F(x)=\int_{4}^{-2}\left(\dot{x}^{2}+4 x^{2}\right) d t \rightarrow \text { extr }, \\ x(-4)=16, \quad x(-2)=4 . \end{array} \]

4.6 Variational calculus

3.62 $

Problem: Let \( M=C[a ; b]- \) be the set of all continuous functions \( y(x) \), given on the interval \( [a ; b] \). Given functional \( I[y(x)]=\pi \int_{a}^{b} y^{2}(x) d x \). Select three different functions that belong to \( C[a ; b] \), and obtain the corresponding values of \( I[y(x)] \), by taking specific \( \mathrm{a} \) and \( \mathrm{b} \).

4.7 Variational calculus

1.21 $

Problem: Let \( M=C[-1 ; 1] \) - the class of the functions \( y(x) \), continuous on segment \( [1 ; 1] \). Given functional \( I[y(x)]=\int_{-1}^{1} \varphi(x, y) d x \), where \( \varphi(x, y) \) - is the function, defined and continuous for all \( x \in[-1 ; 1] \) and real \( y \). For the given function \( \varphi(x, y) \) select couple of functions \( y(x) \) and find the corresponding values of the functional. \[ \varphi=\frac{x}{2-y^{2}} \text {. } \]

4.8 Variational calculus

1.21 $

Problem: Let \( M=C_{1}[3 ; 5] \) - the class of the functions \( y(x) \), having a continuous derivative on the interval [3;5], and let \( I[y(x)]=\left(y^{\prime}\left(x_{0}\right)\right)^{3}+y^{3}\left(x_{0}\right) \), where \( x_{0} \in[3 ; 5] \). Select couple of functions \( y(x) \) and a point \( x_{0} \in[3 ; 5] \); then find the value of the functional for the taken function and point.

4.9 Variational calculus

0.72 $

Problem: Find the distance between the curves on the specified segment: \[ f(x)=e^{-x^{2}}, \quad f_{1}(x)=0, \quad[-1,2] . \]

4.10 Variational calculus

0.72 $

Problem: For the functional find the variation in the corresponding spaces in the sense of the second definition. \[ I[y(x)]=\int_{a}^{b}\left(x y+y^{\prime 2}\right) d x \]

4.11 Variational calculus

1.69 $

Problem: Find the extremals of the functional: \[ \begin{array}{l} I[y(x)]=\int_{-1}^{1} \sqrt{y\left(1+y^{2}\right)} d x, \\ y(0)=\frac{1}{\sqrt{2}}, \quad y(1)=\frac{1}{\sqrt{2}} . \end{array} \]

4.12 Variational calculus

4.1 $

Problem: Check the validity of the Jacobi condition: \[ \begin{array}{l} I[y(x)]=\int_{0}^{a}\left(y^{2}-y^{2}\right) d x, \quad a>0, \\ a \neq \pi k, \quad y(0)=0, \quad y(a)=0 \end{array} \]

4.13 Variational calculus

4.1 $

Problem: Investigate the functional to an extremum. \[ \begin{array}{l} I[y(x)]=\int_{0}^{\frac{1}{2}}\left(2 y y^{\prime}+y^{\prime 2}-16 y^{2}\right) d x, \\ y(0)=0, \quad y\left(\frac{1}{2}\right)=0 . \end{array} \]

4.14 Variational calculus

2.9 $

Problem: Investigate the functional to an extremum. \[ \begin{array}{l} I[y(x)]=\int_{0}^{1} e^{x}\left(1+y^{2}+\frac{y^{\prime 2}}{2}\right) d x, \\ y(0)=1, \quad y(1)=e \end{array} \]

4.15 Variational calculus

2.9 $

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