Answer to Question #271457 in Differential Equations for Lynil

Question

Determine the order of the given differential equation; also state whether the equation is linear or nonlinear.

1. t²(d²y/dt²)+t(dy/dt)+2y=sint
2. (1+y²)(d²y/dt²)+t(dy/dt)+y=e^t

Answer 1

Solution;

(1)

Given;

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

The highest derivative present in the give differential equation is $\frac{d^2y}{dt^2}$

Therefore the order is 2.

A differential equation is linear if the dependent variable y and its various derivatives have degree one or zero the equation.

Highest power of the highest order derivative is 1.

Therefore the degree is 1.

It's a linear equation.

(2)

Given;

$(1+y^2)\frac{d^2y}{t^2}+t(\frac{dy}{dt})+y=e^t$

Highest derivative is $\frac{d^2y}{dt^2}$ .

Therefore order is 2

The equation has $y^2\frac{d^2y}{dt^2}$ which is a non linear term.

Hence is a non linear differential equation.