Question

1. Give an example of a 3rd order nonlinear ordinary differential equation.

Answer 1

The third order non linear orfinary differential equation is

Y"'(t) + a(Y"(t))^2+ b(Y'(t))^3

To solve this eqn

Substitute z=y′z=y′

z′′(t)+a(z′(t))2+bz3=0z″(t)+a(z′(t))2+bz3=0

Substitute p=z′p=z′

dpdzp+ap2+bz3=0dpdzp+ap2+bz3=0

12(p2)′+ap2+bz3=012(p2)′+ap2+bz3=0

Finally substitute w=p2w=p2

12w′+aw+bz3=012w′+aw+bz3=0

Bernouilli's equationAs a more general solution, if you have an equation of the form

x′′(t)+a(x(t))x′(t)2+b(x(t))=0x″(t)+a(x(t))x′(t)2+b(x(t))=0

then you can make the substitution f(x)=x′(t)2f(x)=x′(t)2 to arrive at the equation

12f′(x)+a(x)f(x)+b(x)=012f′(x)+a(x)f(x)+b(x)=0

Letting μ(x)=exp[∫a(x)dx]μ(x)=exp⁡[∫a(x)dx], we can solve for f(x)f(x):

f(x)=μ(x)−1(C1−2∫μ(x)b(x)dx)f(x)=μ(x)−1(C1−2∫μ(x)b(x)dx)

which can be substituted back for x(t)x(t):

x′=μ(x)−1/2(C1−2∫μ(x)b(x)dx)1/2x′=μ(x)−1/2(C1−2∫μ(x)b(x)dx)1/2

and solved implicitly:

C2+t−∫[μ(x)(C1−2∫μ(x)b(x)dx)−1]1/2dx=0

Some more examples of 3rd order non linear eqn are

Y"' = aY^5/2+ bY^7/2 etc