2 Applying the method of separation of variables to ODEs

Example 3

Use the method of separation of variables to solve the differential equation

$\frac{dy}{dx}=\frac{3x^2}{y}$

Solution

The equation already has the form

$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$

where

$f\left(x\right)=3x^2\text{and}g\left(y\right)=1∕y.$

Dividing both sides by $g\left(y\right)$ we find

$y\frac{dy}{dx}=3x^2$

Integrating both sides with respect to $x$ gives

$\int \; <!--nolimits\; -->y\frac{dy}{dx}dx=\int \; <!--nolimits\; -->3x^{2}dx$

that is

$\int \; <!--nolimits\; -->ydy=\int \; <!--nolimits\; -->3x^{2}dx$

Note that the left-hand side is an integral involving just $y$ ; the right-hand side is an integral involving just $x$ . After integrating both sides with respect to the stated variables we find

$\frac{1}{2}{y}^2=x^3+c$

where $c$ is a constant of integration. (You might think that there would be a constant on the left-hand side too. You are quite right but the two constants can be combined into a single constant and so we need only write one.)

We now have a relationship between $y$ and $x$ as required. Often it is sufficient to leave your answer in this form but you may also be required to obtain an explicit relation for $y$ in terms of $x$ . In this particular case

$y^2=2x^3+2c$

so that

$y=\pm \sqrt{2x^3+2c}$

Task!

Use the method of separation of variables to solve the differential equation

$\frac{dy}{dx}=\frac{cosx}{sin2y}$

First separate the variables so that terms involving $y$ and $\frac{dy}{dx}$ appear on the left, and terms involving $x$ appear on the right:

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Exercises

1. Solve the equation

   $\frac{dy}{dx}=\frac{{\text{e}}^{-x}}{y}.$

2. Solve the following equation subject to the condition $y\left(0\right)=1$ :

   $\frac{dy}{dx}=3x^2{\text{e}}^{-y}$

3. Find the general solution of the following equations:
   1. $\frac{dy}{dx}=3$ ,
   2. $\frac{dy}{dx}=\frac{6sinx}{y}$
4. 1. Find the general solution of the equation

      $\frac{dx}{dt}=t\left(x-2\right).$

   2. Find the particular solution which satisfies the condition $x\left(0\right)=5$ .
5. Some equations which do not appear to be separable can be made so by means of a suitable

   substitution. By means of the substitution $z=y∕x$ solve the equation

   $\frac{dy}{dx}=\frac{y^2}{x^2}+\frac{y}{x}+1$

6. The equation

   $iR+L\frac{di}{dt}=E$

   where $R$ , $L$ and $E$ are constants arises in electrical circuit theory. This equation can be

   solved by separation of variables. Find the solution which satisfies the condition $i\left(0\right)=0$ .

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