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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

#### Solution Preview

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 = âˆ‚2z/âˆ‚x2 , s = âˆ‚2z/âˆ‚xâˆ‚y, t = âˆ‚2z/âˆ‚y2
(c) (2D2 + 5DDâ€² + 2Dâ€²2)z = 0 (d) âˆ‚3z/âˆ‚x3 - 3âˆ‚3z/âˆ‚x2âˆ‚y + 2âˆ‚3z/âˆ‚xâˆ‚y2 = 0

Solution :- (a) r = a2t => D2z = a2Dâ€²z => ( D2 - a2 Dâ€²2 )z =0 -------------------(1)
where D â‰¡ âˆ‚/âˆ‚x , Dâ€² â‰¡ ...

#### 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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