Question #107597
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
1
Expert's answer
2020-04-02T13:20:01-0400

a)


zxy=6x3yz_{xy}=6x^3y

zx=3x3y2+f(x)z_x=3x^3y^2+f(x)

The general solution of the partial differential equation. 

z=34x4y2+F(x)+G(y)z={3\over 4}x^4y^2+F(x)+G(y)

b)

z(x,0)=6x2=>F(x)+G(0)=6x2=>z(x,0)=6x^2=>F(x)+G(0)=6x^2=>

=>F(x)=6x2,G(0)=0=>F(x)=6x^2, G(0)=0

z(1,y)=6cosy=>34y2+6+G(y)=6cosy=>z(1,y)=6\cos{y}=>{3\over 4}y^2+6+G(y)=6\cos{y}=>

=>G(y)=34y2+6cosy6=>G(y)=-{3 \over 4}y^2+6\cos{y}-6

z=34x4y2+6x234y2+6cosy6z^*={3\over 4}x^4y^2+6x^2-{3 \over 4}y^2+6\cos{y}-6


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Comments

Assignment Expert
07.04.20, 15:22

In part b), the solution states F(x)+G(0)=6x^2. The G(0) is equal to the specific number. If F(x)=6x^2, then G(0)=0. If G(0)=a, where a is not equal to zero, then F(x)=6x^2-a. Nevertheless, this choice does not change the final expression of the particular solution.

Vishal Ramjattan
07.04.20, 06:56

Is G(0) equal to 0 because there is no y term on the other side of the equation? ( 2nd line of part b)

Assignment Expert
02.04.20, 20:45

The position of 6 was not specified in the comment.

Rehan sudama
02.04.20, 20:42

Should the 6 be positive?

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