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Magazine Problem 23
Discuss the extremum of the functional \(J[y]=\int_{x_{0}}^{x_{1}}\left(y^{2}+2 x y y^{\prime}\right) \mathrm{d} x\), the boundary conditions are \(y\left(x_{0}\right)=y_{0}, y\left(x_{1}\right)=y_{1}\).
Problem 24
Discuss the extremum of the functional \(J[y]=\int_{0}^{1}\left(x y+y^{2}-2 y^{2} y^{\prime}\right) \mathrm{d} x\), the boundary conditions are \(y(0)=1, y(1)=2\).
Problem 25
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}} y^{\prime}\left(1+x^{2} y^{\prime}\right) \mathrm{d} x\).
Problem 26
Find the extremal curve of the functional \(J[y]=\int_{1}^{2} x^{2} y^{\prime 2} \mathrm{~d} x\), the boundary conditions are \(y(1)=1, y(2)=\frac{1}{2}\).
Problem 27
Find the extremal curve of the functional \(J[y]=\int_{0}^{1}\left(x+y^{\prime 2}\right) \mathrm{d} x\), the boundary conditions are \(y(0)=1, y(1)=2\).
Problem 29
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}}\left(x y^{\prime}+y^{\prime 2}\right) \mathrm{d} x\).
Problem 30
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}} x^{n} y^{\prime 2} \mathrm{~d} x\), the boundary conditions are \(y\left(x_{0}\right)=y_{0}, y\left(x_{1}\right)=y_{1}\).
Problem 31
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}} \frac{y^{2}}{x^{k}} \mathrm{~d} x\), where, \(k>0\).
Problem 33
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}} \frac{x}{x+y^{\prime}} \mathrm{d} x\).
Problem 34
Find the extremal curve of the functional \(J[y]=\int_{x_{0}}^{x_{1}} \frac{\sqrt{1+y^{\prime 2}}}{x+k} \mathrm{~d} x\), where, \(k\) is a constant.
