# Linear Homogeneous Partial Differential Equation

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

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

#### Solution Preview

Please see the attached file.

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.