Chapter 1.3, Problem 1P

In each of Problems $1$ through $6$, determine the order of thegiven differential equation; also state whether the equation islinear or nonlinear.

$t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$

To determine

The order of the differential equation $t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$, and whether it is linear or not.

Answer to Problem 1P

Solution:

The differential equation $t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$ is of the second order linear differential equation.

Explanation of Solution

Given information:

The differential equation $t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$.

The order of the highest derivative in an ordinary or partial differential equation is called the order of the differential equation.

For $t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$, the highest order is $2$.

The given differential equation is, therefore, of the second order.

An $n$ th orderordinary differential equation of the form $a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdot \cdot \cdot +a_n(t)y=g(t)$ is called linear. Where the functions $a_0,a_1,\mathrm{...},a_n$ called the coefficients of the equation and they are dependent on $t$. The ordinary differential equation which is not in the above form is non-linear differential equation.

In the differential equation $t^2\frac{d^{2}y}{d{t}^2}+t\frac{dy}{dt}+2y=\sin t$, $n=2$, the coefficients $a_0(t)=t^2,a_1(t)=t,a_2(t)=2$ are dependent on $t$ and $g(t)=\sin t$.

Therefore, the given differential equation islinear.