Question

By calculating partial derivatives, test each of the following equality:

Help me do the "b" in the picture.

Answer 1

the equation is satisfied by the expression for z

Explanation:

Given z = y/(y² -a²x²) you want to show that it satisfies a second-order partial derivative equation.

Partial derivatives

`(\partial z)/(\partial x)=2a^2xy(y^2-a^2x^2)^(-2)\\\\(\partial^2 z)/(\partial x^2)=(2a^2y)((y^2-a^2x^2)^2-2x(y^2-a^2x^2)(-2a^2x))/((y^2-a^2x^2)^4)=(2a^2y(y^2+3a^2x^2))/((y^2-a^2x^2)^3)`

and ...

`(\partial z)/(\partial y)=((y^2-a^2x^2)-y(2y))/((y^2-a^2x^2)^2)=-(y^2+a^2x^2)/((y^2-a^2x^2)^2)\\\\(\partial ^2z)/(\partial y^2)=-((y^2-a^2x^2)^2(2y)-(y^2+a^2x^2)(2(y^2-a^2x^2)(2y)))/((y^2-a^2x^2)^4)=(2y(y^2+3a^2x^2))/((y^2-a^2x^2)^3)`

We can define ...

q = 2y(y²+3a²x²)/(y²-a²x²)³

so the equation of (b) becomes ...

(a²q) - a²(q) = 0 . . . . . . true

The given expression for z satisfies the partial differential equation.

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Additional comment

We used the power rule and the quotient rule for derivatives. Tedious, but not difficult.