Find out the answer to "Differentiation question - (an application of similarity solutions)ψ=x13f(η)where,η=yx23"

High School
Calculus and Analysis
Calculus 2
Differential equations

Emmy Decker

Answered question

2022-04-12

Differentiation question - (an application of similarity solutions)

ψ = x^{\frac{1}{3}} f (η)

where,

η = \frac{y}{x^{\frac{2}{3}}}

Answer & Explanation

shulsellsmine2aaf

Beginner2022-04-13Added 5 answers

Step 1
The derivative
$\frac{\partial ψ}{\partial y} = x^{- 1 / 3} \frac{d f}{d η}$
Then we get:
$\frac{\partial^{2} ψ}{\partial y^{2}} = x^{- 1} \frac{d^{2} f}{d η^{2}}$
If we want to determine $\frac{\partial^{2} ψ}{\partial x \partial y}$ we have to use the product rule, which should give:
$\frac{\partial^{2} ψ}{\partial x \partial y}$
$= \frac{\partial x^{- 1 / 3}}{\partial x} \frac{d f}{d η} + x^{- 1 / 3} \frac{\partial}{\partial x} \frac{d f}{d η}$
$= - \frac{2 y \frac{d^{2} f}{d η^{2}} + x^{2 / 3} \frac{d f}{d η}}{3 x^{2}}$

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