Differential Equation - Practice Questions & MCQ

Consider the following examples,

$
\begin{aligned}
& \mathrm{x}^2+\mathrm{x}+1=0 \\
& \sin x+\cos x=0 \\
& \mathrm{x}+\mathrm{y}=9 \\
& \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y}=0
\end{aligned}
$

We are already familiar with the equations of the type (1), (2) and (3) but equation (4) is new for us. An equation of the form (4) is known as a differential equation. In this chapter, we will study some basic concepts related to the differential equations.

Observe that equations (1), (2), and (3) involve independent and/or dependent variables (variables) only but equation (4) involves variables as well as the derivative of the dependent variable y with respect to the independent variable x. Such an equation is called a differential equation.

In general, an equation involving derivatives of the dependent variable with respect to independent variables is called a differential equation.

The following relations are some of the examples of differential equations:

(i) $\frac{d y}{d x}=\sin 2 x+\cos x$
(ii) $k \frac{\mathrm{~d}^2 \mathrm{y}}{\mathrm{dx}^2}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^2\right]^{3 / 2}$

Now, a differential equation that involves derivatives with respect to a single independent variable is known as an ordinary differential equation.

For example

(i) $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\sin \mathrm{x}$
(ii) $\frac{\mathrm{d}^3 \mathrm{y}}{\mathrm{dx}^3}+2 \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=\mathrm{e}^{\mathrm{x}}$

In this chapter, we shall confine ourselves to the study of ordinary differential equations only.

NOTE : 

From now on, we will use the term 'differential equation' or 'DE' for 'ordinary differential equation'.
We will use the following notations for derivatives:
$
\frac{d y}{d x}=y^{\prime}, \frac{d^2 y}{d x^2}=y^{\prime \prime}, \frac{d^3 y}{d x^3}=y^{\prime \prime \prime}
$
For derivatives of higher order, we use the notation $\mathrm{y}_{\mathrm{n}}$ for $\mathrm{n}^{\text {th }}$ order derivative $\frac{d^n y}{d x^n}$.

Order and Degree of a Differential Equation

Order

The order of a differential equation is the highest order of any derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

Consider the following example:

(i) $\frac{\mathrm{dy}}{\mathrm{dx}}=\sin 2 \mathrm{x}+\cos \mathrm{x}$ is of order 1
(ii) $\frac{d^2 y}{d x^2}+y=0$ is of order 2
(iii) $\frac{d^3 y}{d x^3}+2\left(\frac{d^2 y}{d x^2}\right)^5+y=e^x$ is of order 3

Degree

Consider the following example:

$
\begin{aligned}
& \frac{d^3 y}{d x^3}+2\left(\frac{d^2 y}{d x^2}\right)^2+\frac{d y}{d x}=x y^2 \\
& \frac{d^3 y}{d x^3}+\sin \frac{d y}{d x}=0
\end{aligned}
$

Observe that equation (i) is a polynomial equation in derivatives:  y''', y'', and y' in this case: these terms occur only with some whole number power. So for this equation, the degree of differential equation can be defined. 

But equation (ii) is not a polynomial in y’, as sin(y') term is present , so degree of this equation is not defined.

Now, consider another example,

$
\left(\frac{d^2 y}{d x^2}\right)^{\frac{1}{2}}=\left(y+\left(\frac{d y}{d x}\right)^4\right)^{\frac{1}{3}}
$

Here, derivatives have no integral power. We can rewrite the equation as
$
\left(\frac{d^2 y}{d x^2}\right)^3=\left(y+\left(\frac{d y}{d x}\right)^4\right)^2
$

After the expansion of these brackets, this will become a polynomial equation of derivatives. So, its degree is defined.

The degree of differential equation is the degree (power) of the highest order derivative present in the equation after the differential equation has been made free from the radicals and fractions as far as the derivatives are concerned.

So, the degree of (i) is 1, as the power of the highest order derivative (y''') is 1.

NOTE

Order and degree (if defined) of a differential equation are always positive integers.