Answer to Question #231169 in Differential Equations for Randal Rodriguez

Question

determine the order and the degree of the given

differential equation; also state whether the equation is linear or nonlinear.

2.\left(1\:+\:y^2\right)\frac{d^4y}{dt^4}\:+t\:\frac{dy}{dt}+\:y\:=\:e^t

Answer 1

$$(1+y^2)\dfrac{d^4y}{dt^4}+t\dfrac{dy}{dt}+y=e^t$$

The highest order derivative present in the differential equation is the order of the differential equation.

Degree is the highest power of the highest order derivative in the differential equation, after the equation has been cleared from fractions and the radicals as for as the derivatives are concerned.

Here the highest order derivatives is $\dfrac{d^4y}{dt^4}.$ So the order of the differential equation is $4.$

Power of the highest order derivative $\dfrac{d^4y}{dt^4}$ is $1.$ Degree $=1.$

An nth order differential equation is said to be linear if it satisfies the following two conditions:

(1) the dependent variable (y) and all its derivatives in the equation are of power one.

(2) all the coefficients and the function are either constants or depend only on the independent variable (t). 

If any one of these 2 conditions is not satisfied, then the DE is said to be nonlinear DE. 

Therefore the given differential equation is nonlinear.