Chapter 4: Q16MP (page 239)

If $z = \frac{1}{x} f (\frac{y}{x})$ , prove that $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$ .

Short Answer

Expert verified

It is proved that the equation $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$ $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$ .

Step by step solution

Given Information

The given information is $z = \frac{1}{x} f (\frac{y}{x})$ .

Find the partial differentiation ∂z∂xand ∂z∂y .

Calculate the partial differentiation with respect to $x$ as:

$\frac{鈭� z}{鈭� x} = \frac{1}{x} f' (\frac{y}{x}) (\frac{- y}{x^{2}}) - \frac{1}{x^{2}} f (\frac{y}{x}) \frac{鈭� z}{鈭� x} = \frac{- y}{x^{3}} f' (\frac{y}{x}) - \frac{1}{x^{2}} f (\frac{y}{x})$

Now calculate the partial differentiation with respect to $y$ as:

$\frac{鈭� z}{鈭� y} = \frac{1}{x} f' (\frac{y}{x}) (\frac{1}{x}) + f (\frac{y}{x}) (0) \frac{鈭� z}{鈭� y} = \frac{1}{x^{2}} f' (\frac{y}{x})$

Perform the required multiplication.

Multiply $\frac{鈭� z}{鈭� x}$ by $x$ as:

$x \frac{鈭� z}{鈭� x} = x (\frac{- y}{x^{3}} f' (\frac{y}{x}) - \frac{1}{x^{2}} f (\frac{y}{x})) x \frac{鈭� z}{鈭� x} = \frac{- y}{x^{2}} f' (\frac{y}{x}) - \frac{1}{x} f (\frac{y}{x})$

Now multiply $\frac{鈭� z}{鈭� y}$ by as:

$y \frac{鈭� z}{鈭� y} = \frac{y}{x^{2}} f' (\frac{y}{x})$

Now adding $x \frac{鈭� z}{鈭� x} = \frac{- y}{x^{2}} f' (\frac{y}{x}) - \frac{1}{x} f (\frac{y}{x})$ and $y \frac{鈭� z}{鈭� y} = \frac{y}{x^{2}} f' (\frac{y}{x})$ as:

$x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = \frac{- y}{x^{2}} f' (\frac{y}{x}) - \frac{1}{x} f (\frac{y}{x}) + \frac{y}{x^{2}} f' (\frac{y}{x}) + \frac{1}{x} f (\frac{y}{x}) x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = \frac{- y}{x^{2}} f' (\frac{y}{x}) + \frac{y}{x^{2}} f' (\frac{y}{x}) - \frac{1}{x} f (\frac{y}{x}) + \frac{1}{x} f (\frac{y}{x}) x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$

Hence, proved that the equation . $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$

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If $z = \frac{1}{x} f (\frac{y}{x})$ , prove that $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$ .

Short Answer

Step by step solution

Given Information

Find the partial differentiation ∂z∂xand ∂z∂y .

Perform the required multiplication.

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91影视

If z=1xf(yx), prove that x鈭�z鈭�x+y鈭�z鈭�y+z=0.

Short Answer

Step by step solution

Given Information

Find the partial differentiation ∂z∂xand ∂z∂y .

Perform the required multiplication.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Mechanics and Materials

Further Mechanics and Thermal Physics

Fundamentals of Physics

Astrophysics

Famous Physicists

Study anywhere. Anytime. Across all devices.

If $z = \frac{1}{x} f (\frac{y}{x})$ , prove that $x \frac{鈭� z}{鈭� x} + y \frac{鈭� z}{鈭� y} + z = 0$ .