3 Classifying differential equations

When solving differential equations (either analytically or numerically) it is important to be able to recognise the various kinds that can arise. We therefore need to introduce some terminology which will help us to distinguish one kind of differential equation from another.

• An ordinary differential equation (ODE) is any relation between a function of a single variable and its derivatives. (All differential equations studied in this workbook are ordinary.)
• The order of a differential equation is the order of the highest derivative in the equation.
• A differential equation is linear if the dependent variable and its derivatives occur to the first power only and if there are no products involving the dependent variable or its derivatives.

Example 2

Classify the differential equations specifying the order and type (linear/non-linear)

1. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=x^2$
2. $\frac{d^{2}x}{d{t}^2}={\left(\frac{dx}{dt}\right)}^3+3x$
3. $\frac{dx}{dt}-x=t^2$
4. $\frac{dy}{dt}+cosy=0$
5. $\frac{dy}{dt}+y^2=4$

Solution

1. Second order, linear.
2. Second order, non-linear (because of the cubic term).
3. First order, linear.
4. First order, non-linear (because of the $cosy$ term).
5. First order, non-linear (because of the $y^2$ term).

Note that in (1) the independent variable is $x$ whereas in the other cases it is $t$ .

In (1), (4) and (5) the dependent variable is $y$ and in (2) and (3) it is $x$ .

Exercises

1. In this RL circuit the switch is closed at $t=0$ and a constant voltage $E$ is applied.

   

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   The voltage across the resistor is $iR$ where $i$ is the current flowing in the circuit and $R$ is the (constant) resistance. The voltage across the inductance is $L\frac{di}{dt}$ where $L$ is the constant inductance.

   Kirchhoff’s law of voltages states that the applied voltage is the sum of the other voltages in the circuit. Write down a differential equation for the current $i$ and state the initial condition.

2. The diagram below shows the graph of $i$ against $t$ (from Exercise 1). What information does this graph convey?

   

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3. In the LCR circuit below the voltage across the capacitor is $q∕C$ where $q$ is the charge on the capacitor, and $C$ is the capacitance. Note that $\frac{dq}{dt}=i$ . Find a differential equation for $i$ and write down the initial conditions if the initial charge is zero and the switch is closed at $t=0$ .

   

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4. Find differential equations satisfied by
   1. $y=Acos4x+Bsin4x$
   2. $x=A{\text{e}}^{-2t}$
   3. $y=Asinx+Bsinhx+Ccosx+Dcoshx$ (harder)
5. Find the family of solutions of the differential equation $\frac{dy}{dx}=-2x.$  Sketch the curves of some members of the family on the same axes. What is the solution if $y\left(1\right)=3$ ?
6. 1. Find the general solution of the differential equation $y^{″}=12x^2.$
   2. Find the solution which satisfies $y\left(0\right)=2,y\left(1\right)=8$
   3. Find the solution which satisfies $y\left(0\right)=1,y^{\prime }\left(0\right)=-2.$
7. Classify the differential equations
   1. $\frac{d^{2}x}{d{t}^2}+3\frac{dx}{dt}=x$
   2. $\frac{d^{3}y}{d{x}^3}={\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}$
   3. $\frac{dy}{dx}+y=sinx$
   4. $\frac{d^{2}y}{d{x}^2}+y\frac{dy}{dx}=2$ .

Answer