# Partial Differential Equation and Solutions (Essay Sample)

PARTIAL DIFFERENTIAL EQUTION AND SOLUTIONS

1. Solve given

Solution

First the equation can be expressed as;

Then, the general solution of the Partial Differential Equation is;

To find a particular solution of the given equation, we assume that

Where is known constant. This is the method of undermined coefficient as in ODE

Thus, we have;

Putting in the given equation, we get;

which on comprising the coefficient, gives;

Hence the required general solution is

2. Find the complete solution of

Solution

complete solution of , in this expression, we involve differentiate partially, that is rather computation using partial differential equation.

PARTIAL DIFFERENTIAL EQUTION AND SOLUTIONS

1 Solve ∂2U∂x2=∂2U∂Y2 given Uxy=XxY(y)

Solution

First the equation can be expressed as;

∂2U∂x2-∂2U∂Y2=0

Then, the general solution of the Partial Differential Equation is;

U=Fx+iy+G(x-iy)

To find a particular solution of the given equation, we assume that U=αXxY(y)

Where α is known constant. This is the method of undermined coefficient as in ODE

Thus, we have;

∂2U∂x2=αXxYy.αYy

∂2U∂Y2=αXxYy.αXx

Putting in the given equation, we get;

αYy ∂2U∂x2+αXx∂2U∂Y2 which on comprising the coefficient, gives;

2α=XxYy=Uxy

Hence the required general solution is

U=Fx+iy+Gx-iy+2XxY(y

2 Find the complete solution of y2zp-x2zq=x2y

S

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