Solving differential equations

In a nutshell

Differential equations are equations that involve a function and its derivative. It is possible to solve a very specific type of differential equation (called "separable differential equations") using a method called "separation of variables".

Separable differential equations

A separable differential equation is an equation of the form:

​$$\dfrac{dy}{dx}=f(x)g(y)$$​​

where $$f(x)$$ and $$g(y)$$ are given functions.

Separation of variables

Separation of variables is the technique used to solve separable differential equations. The process is as follows.

procedure

Note: Here you are treating $$dy$$ and $$dx$$ as if they are themselves variables. While this is not quite true, you can manipulate them as such.

Example 1

Solve the differential equation $$\dfrac{dy}{dx}=xy$$.

"Separate the variables", so divide both sides by $$y$$ and multiply both sides by $$dx$$:

​$$\begin{aligned} \dfrac{dy}{dx}&=xy\\ \Rightarrow \dfrac1y \,\dfrac{dy}{dx}&=x\\ \Rightarrow \dfrac1y \, dy &= x \, dx \end{aligned}$$​​

Integrate both sides:

$$\begin{aligned}\dfrac1y \, dy &= x \, dx \\ \int \dfrac1y \, dy &= \int x \, dx \\ \ln\vert y \vert &= \dfrac12 x^2+C\end{aligned}$$​​

Rearrange for $$y$$:

​$$\begin{aligned} \ln\vert y \vert &= \dfrac12 x^2+C\\y&=e^{\frac12 x^2+C}\\&=e^Ce^{\frac12 x^2}\\&=Ae^{\frac12 x^2}, \ where \ A=e^c \end{aligned}$$​​

​$$\underline{y=Ae^{\frac12 x^2}}$$​​

Note: It is perfectly fine to replace a constant (in this case $$e^c$$) with a simpler constant (such as $$A$$). The are both just constants. 

General and particular solutions

The solution to the above example is a general solution to the differential equation. This is because there are an infinite number of unique functions for $$y$$, as any value of $$A$$ can be chosen.

A particular solution is a solution that has no arbitrary constants - it is the only solution satisfying the differential equation and an added boundary condition $$(a,b)$$. This means that when $$x=a$$​, $$y=b$$​ and hence the constant of integration can be found.

Example 2

Find the particular solution of the differential equation $$\dfrac{dy}{dx}=\dfrac{\tan(y)}{x}$$ that passes through the point $$(1,\dfrac{\pi}{2})$$.

​​

Separate the variables by multiplying both sides by $$dx$$ and dividing both sides by $$\tan(y)$$:

​$$\begin{aligned} \dfrac{dy}{dx}&=\dfrac{\tan(y)}{x}\\\dfrac{1}{\tan(y)}\,dy&= \dfrac{1}{x}\,dx\\ \cot(y)\,dy&=\dfrac1x \,dx \end{aligned}$$​​

Integrate both sides:

​$$\begin{aligned} \cot(y)\,dy&=\dfrac1x \,dx \\ \int \cot(y)\,dy&=\int \dfrac1x \,dx \\\ln \vert\sin(y)\vert &=\ln \vert x \vert+C \end{aligned}$$​​

Use the boundary condition:

​$$\begin{aligned}\ln \vert\sin(y)\vert &=\ln \vert x \vert+C \\x=1&, y=\dfrac{\pi}{2}\\ \ln\left\vert \sin\left(\dfrac{\pi}{2}\right) \right\vert&=\ln \vert1 \vert+C\\ \ln(1)&=\ln(1)+C\\C&=0 \end{aligned}$$​​

Rewrite the solution with the solved value of $$C$$ and rearrange:

​$$\begin{aligned}\ln \vert \sin(y)\vert &= \ln(\vert x\vert)\\ \vert \sin(y)\vert&= x \end{aligned}$$​​

​$$\underline{x=\vert\sin(y)\vert}$$​​

Note: This example also showcases that, unless otherwise instructed, it is not necessary for the solution to be written in the form $$y=f(x)$$.