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Partial Differential Equations : Linear & Quasi-Linear PDEs

Study concepts, example questions & explanations for Partial Differential Equations

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Example Question #3 : Partial Differential Equations

Determine if the statement is true or false:

If the $\displaystyle z$-axis is the axis of symmetry and a surface is revolving around it and $\displaystyle f$ is an arbitrary function, then the partial differential equation associated with that said surface, satisfies the equation:

$\displaystyle z=f(x^2+y^2)$

Possible Answers:

False

True

Correct answer:

True

Explanation:

To determine the truth of this statement, assume the following.

$\displaystyle f$ is some function

$\displaystyle u=x^2+y^2$

From here, differential $\displaystyle u$ with respect to $\displaystyle x$ and $\displaystyle y$.

$\displaystyle z=f(x^2+y^2)$

$\\\frac{dz}{dx}=2x\cdot f'(u) \\\\\frac{dz}{dy}=2y\cdot f'(u)$ $\displaystyle \\\frac{dz}{dx}=2x\cdot f'(u) \\\\\frac{dz}{dy}=2y\cdot f'(u)$

Next eliminate f'(u) $\displaystyle f'(u)$ as it is an arbitrary function.

This leads to the result,

$y\cdot \frac{dz}{dx}-x\cdot \frac{dy}{dz}=0$ $\displaystyle y\cdot \frac{dz}{dx}-x\cdot \frac{dy}{dz}=0$

Therefore, the statement,

$\displaystyle z=f(x^2+y^2)$

is true.

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Partial Differential Equations : Linear & Quasi-Linear PDEs

Study concepts, example questions & explanations for Partial Differential Equations

All Partial Differential Equations Resources

Example Questions

Example Question #3 : Partial Differential Equations

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