Partial Differential Equation and Solutions (Essay Sample)

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