Flashcards
2022-01-19
What’s the reducible differential equations topic about??
Transforming complicated differential equations into simpler ones using a substitution.
If $z = \frac{y}{x}$, or $y = xz$ what is $\frac{\text{d}y}{\text{d}x}$??
$$ \frac{\text{d}y}{\text{d}x} = z + x\frac{\text{d}z}{\text{d}x} $$
What’s the first stage in doing a first-order reducible differential equations question??
Working out how to substitute the derivative by differentiating.
If a reducible differential equations question asks you to substitute $z = \frac{y}{x}$ then what is the first step??
Rearranging to
$$ y = zt $$
Then differentiating
$$ \frac{\text{d}y}{\frac{d}t} = z + t\frac{\text{d}z}{\text{d}t} $$
If $z = \frac{1}{y^2}$, or $y = z^{-1/2}$ then what is $\frac{\text{d}y}{\text{d}x}$??
$$ \frac{\text{d}y}{\text{d}x} = -\frac{1}{2} z^{-3/2} \frac{\text{d}z}{\text{d}x} $$
2022-01-20
What is the aim of any reducible differential equations question??
Making sure that all instances of the variable (including derivatives) have been replaced.
If you’re asked to do a substitution for removing $x$ using a function of $z$ in a second differential equation, what would $\frac{\text{d}y}{\text{d}x}$ need to become??
$$ \frac{\text{d}y}{\text{d}z} $$
If you’re asked to do a substitution for removing $x$ using a function of $z$ in a second differential equation, what would $\frac{\text{d}^2y}{\text{d}x^2}$ need to become??
$$ \frac{\text{d}^2y}{\text{d}z^2} $$
Imagine you’ve been given a reducible second-order differential equation where $x = e^u$ and you need a solution for $y$. How could you work out $\frac{\text{d}y}{\text{d}x}$ given that $y$ never appears in the substitution??
Use the chain rule but backwards
$$ \frac{\text{d}y}{\text{d}x} = \frac{\text{d}y}{\text{d}u} \frac{\text{d}z}{\text{d}u} $$
When completing a second-order reducible differential equations question, what is it useful to do at every step where you come up with a formula for a derivative (i.e. $\frac{\text{d}u}{\text{d}x} = \frac{1}{x}$ or $\frac{\text{d}y}{\text{d}x} = \frac{1}{x} \frac{\text{d}y}{\text{d}u}$??
Box the result so that its easy to refer back to later.
You’re completing a second-order reducible differential equations question and finding a nice form for $\frac{\text{d}^2y}{\text{d}x^2}$ in terms of $\frac{\text{d}y}{\text{d}u}$ and $\frac{\text{d}^2 y}{\text{d}u^2}$ is hard. What could you try instead to attempt it another way??
Starting off by finding a form for
$$ \frac{\text{d}^2y}{\text{d}u^2} = \frac{\text{d}}{\text{d}u}\left[ \frac{\text{d}y}{\text{d}u} \right] $$
What’s another way of writing $\frac{\text{d}x}{\text{d}t}$ when you’re using a substitution involving a variable $z$??
$$ \frac{\text{d}x}{\text{d}t} = \frac{\text{d}x}{\text{d}z} \times \frac{\text{d}z}{\text{d}t} $$
What is $$\frac{\text{d}}{\text{d}u} (\frac{\text{d}y}{\text{d}x})$$??
$$ \frac{\text{d}^2 y}{\text{d}x^2} \times \frac{\text{d}x}{\text{d}u} $$
2022-01-21
What do you have to remember to do at the end of every reducible differential equations question??
Reverse the substitution.
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date: 2022-01-19 18:44
tags:
- '@?public'
- '@?school'
- '@?further-maths'
- '@?futher-pure-1'
- '@?differential-equations'
title: Further Maths - Reducible Differential Equations