Symmetric solutions of PDEs via ODEs

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Problem 4256 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.15 "Higher-Order Equations Admitting a Reduction of Order"

Find the function u=f(x2+y2)u=f(x^{2}+y^{2}) satisfying Laplace's equation ∂2u∂x2+∂2u∂y2=0\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}=0.

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Problem 4257 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.15 "Higher-Order Equations Admitting a Reduction of Order"

Find the general form of the functions u=f(x2+y2+z2)u=f(x^{2}+y^{2}+z^{2}) satisfying the three-variable Laplace equation ∂2u∂x2+∂2u∂y2+∂2u∂z2=0\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}+\dfrac {\partial^{2}u}{\partial z^{2}}=0.

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Problem 4258 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.15 "Higher-Order Equations Admitting a Reduction of Order"

Find the general form of the functions u=f(x2+y2)u=f\left(\sqrt{x^{2}+y^{2}}\right) satisfying the biharmonic equation

∂4u∂x4+2∂4u∂x2 ∂y2+∂4u∂y4=0. \frac{\partial ^{4}u}{\partial x^{4}}+2\frac{\partial ^{4}u}{\partial x^{2}\, \partial y^{2}}+\frac{\partial ^{4}u}{\partial y^{4}}=0.
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Problem 4259 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.15 "Higher-Order Equations Admitting a Reduction of Order"

Surfaces for which the mean curvature is equal to zero are called minimal. The coordinates of their points satisfy the equation

(1+p2)t−2pqs+(1+q2)r=0, (1+p^{2})t-2pqs+(1+q^{2})r=0,

where

p=∂z∂x,q=∂z∂y,r=∂2z∂x2,s=∂2z∂x ∂y,t=∂2z∂y2. p=\frac{\partial z}{\partial x}, \quad q=\frac{\partial z}{\partial y}, \quad r=\frac{\partial ^{2}z}{\partial x^{2}}, \quad s=\frac{\partial ^{2}z}{\partial x\, \partial y}, \quad t=\frac{\partial ^{2}z}{\partial y^{2}}.

For surfaces of revolution about the axis OzOz one must have z=f(x2+y2)z=f\left(\sqrt{x^{2}+y^{2}}\right). Find the minimal surfaces of revolution. (Monge.)

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References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.