Section 15.2

Reduction of Second-Order Linear Equations to Canonical Form

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Problem 5817

Reduce the following differential equation to canonical form:

∂2u∂x2−∂2u∂y2=0. \frac{\partial ^{2} u}{\partial x^{2}} - \frac{\partial ^{2} u}{\partial y^{2}} = 0.
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Problem 5818

Reduce the following differential equation to canonical form:

x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=0. x^{2}\frac{\partial ^{2} u}{\partial x^{2}}+2xy\frac{\partial ^{2} u}{\partial x \partial y}+y^{2}\frac{\partial ^{2} u}{\partial y^{2}} = 0.
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Problem 5819

Reduce the following differential equation to canonical form:

y2∂2u∂x2+x2∂2u∂y2=0. y^{2}\frac{\partial ^{2} u}{\partial x^{2}}+x^{2}\frac{\partial ^{2} u}{\partial y^{2}} = 0.
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Problem 5820

Reduce the following differential equation to canonical form:

x2∂2u∂x2−y2∂2u∂y2=0. x^{2}\frac{\partial ^{2} u}{\partial x^{2}}-y^{2}\frac{\partial ^{2} u}{\partial y^{2}} = 0.
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Problem 5821

Reduce the following differential equation to canonical form:

(l−x)∂2u∂x2−∂2u∂y2−∂u∂x=0. (l-x)\frac{\partial ^{2} u}{\partial x^{2}} - \frac{\partial ^{2} u}{\partial y^{2}} - \frac{\partial u}{\partial x} = 0.
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Problem 5822

Reduce the following differential equation to canonical form:

(l2−x2)∂2u∂x2−∂2u∂y2−2x∂u∂x−14u=0. (l^{2}-x^{2})\frac{\partial ^{2} u}{\partial x^{2}} - \frac{\partial ^{2} u}{\partial y^{2}} - 2x\frac{\partial u}{\partial x} - \frac{1}{4}u = 0.
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Problem 5823

Find the change of variables under which the equation

∂2u∂x∂y−β′ctg⁡(x−y)∂u∂x+βctg⁡(x−y)∂u∂y+ββ′u=0 \frac{\partial ^{2} u}{\partial x \partial y} - \beta ' \ctg (x-y)\frac{\partial u}{\partial x} + \beta \ctg (x-y)\frac{\partial u}{\partial y} + \beta \beta ' u = 0

is transformed into the equation

∂2v∂ξ∂η−mξ−η∂v∂ξ+nξ−η∂v∂η+p(ξ−η)2v=0. \frac{\partial ^{2} v}{\partial \xi \partial \eta } - \frac{m}{\xi -\eta }\frac{\partial v}{\partial \xi } + \frac{n}{\xi -\eta }\frac{\partial v}{\partial \eta } + \frac{p}{(\xi -\eta )^{2}}v = 0.
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