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# Linear Homogeneous Partial Differential Equation

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Find the General Solution of the equations.

(a) r = a2t
(b) r - 3as + 2a2t = 0 where r = &#8706;2z/&#8706;x2 , s = &#8706;2z/&#8706;x&#8706;y, t = &#8706;2z/&#8706;y2
(c) (2D2 + 5DD&#8242; + 2D&#8242;2)z = 0 (d) &#8706;3z/&#8706;x3 - 3&#8706;3z/&#8706;x2&#8706;y + 2&#8706;3z/&#8706;x&#8706;y2 = 0

https://brainmass.com/math/calculus-and-analysis/linear-homogeneous-partial-differential-equation-23620

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Linear Partial Differential Equation
Linear Homogeneous Partial Differential Equation

Written by :- Thokchom Sarojkumar Sinha

Find the General Solution of the equations.
(a) r = a2t (b) r - 3as + 2a2t = 0 where r = ∂2z/∂x2 , s = ∂2z/∂x∂y, t = ∂2z/∂y2
(c) (2D2 + 5DD′ + 2D′2)z = 0 (d) ∂3z/∂x3 - 3∂3z/∂x2∂y + 2∂3z/∂x∂y2 = 0

Solution :- (a) r = a2t => D2z = a2D′z => ( D2 - a2 D′2 )z =0 -------------------(1)
where D ≡ ∂/∂x , D′ ≡ ...

#### Solution Summary

This solution is comprised of a detailed explanation for finding the general solution of the linear homogeneous partial differential equation.

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