Question

4. (a)Find the general solution of the partial differential equation.

zxy =(6x^3)y


(b) Find the particular solution for which z(x,0) = 6x^2 , z(1, y) = 6cos y

Answer 1

a)

$$z_{xy}=6x^3y$$

$$z_x=3x^3y^2+f(x)$$

The general solution of the partial differential equation. 

$$z={3\over 4}x^4y^2+F(x)+G(y)$$

b)

$$z(x,0)=6x^2=>F(x)+G(0)=6x^2=>$$

$$=>F(x)=6x^2, G(0)=0$$

$$z(1,y)=6\cos{y}=>{3\over 4}y^2+6+G(y)=6\cos{y}=>$$

$$=>G(y)=-{3 \over 4}y^2+6\cos{y}-6$$

$$z^*={3\over 4}x^4y^2+6x^2-{3 \over 4}y^2+6\cos{y}-6$$