Differential Equations: Separation of Variables 1

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Name	Status	Author	Last Modified
Differential Equations: Separation of Variables 1	Ready to use	Ben McGovern	27/02/2023 10:12
Differential Equations: Separation of Variables 2	Ready to use	Ben McGovern	27/02/2023 10:13
Differential Equations: Separation of Variables 3	Ready to use	Ben McGovern	27/02/2023 10:13
Differential Equations: Separation of Variables 4	Ready to use	Ben McGovern	22/09/2022 14:27
Differential Equations: Separation of Variables 5	Ready to use	Ben McGovern	22/09/2022 14:29
Differential Equations: Separation of Variables 6	Ready to use	Ben McGovern	22/09/2022 14:34
Differential Equations: Separation of Variables 7	Ready to use	Ben McGovern	22/09/2022 14:35

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Question statement

Solve the differential equation

$\frac{dy}{dx}=\var{a}xy.$

Give any introductory information the student needs.

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$y=\,$ Gap 0

Advice

If we have a differential equation of the form

$\frac{dy}{dx} = f(x) g(y),$ we are able to find a solution to this equation using the method Separation of Variables.

We can rewrite the above equation in the form

$\frac{1}{g(y)} \frac{dy}{dx} = f(x).$

If we then integrate with respect to $x$ :

$\begin{split} &\int \frac{1}{g(y)} &\frac{dy}{dx} dx = \int f(x) dx, \\\\ \implies &\int \frac{1}{g(y)} &dy = \int f(x) dx. \end{split}$

Integrating both sides, we can write $y$ as a function of $x$ .

Following this method for $\frac{dy}{dx}=\var{a}xy$ :

$\begin{split} &\frac{dy}{dx}=\var{a}xy \\\\ \implies \frac{1}{y}& \frac{dy}{dx} = \var{a}x. \end{split}$

Integrating both sides with respect to $x$ :

$\begin{split} & \int \frac{1}{y} \frac{dy}{dx} dx = \int \var{a}x \, dx, \\ \\ \implies &\int \frac{1}{y} dy = \int \var{a} x \, dx. \end{split}$

Taking the integral of both sides, we are able to find a solution to the differential equation:

$\begin{split}&\quad \int &\frac{1}{y} dy &\,= \int \var{a} x \, dx \\ &\implies &\ln(y) &\,= \simplify{{a}x^2/2} + c, \\ &\implies &\quad y &\,= e^{\simplify{{a}x^2/2+c}}. \end{split}$

Note: You can also have the answer

$y =Ae^{\simplify{{a/2}x^2}},$

where $A=e^c$ . This shows there is still a constant, but indicates it is different to the original constant of integration, $c$ .

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Differential Equations: Separation of Variables 1

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Differential Equations: Separation of Variables 1

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Taxonomy: mathcentre

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Taxonomy: Context

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Ben McGovern 3 years, 8 months ago

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Generated value: `integer`