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

3.6 Ratings (356 Votes)

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 ?(x)=exp[?a(x)dx]?(x)=exp?[?a(x)dx], we can solve for f(x)f(x):

f(x)=?(x)?1(C1?2??(x)b(x)dx)f(x)=?(x)?1(C1?2??(x)b(x)dx)

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

x?=?(x)?1/2(C1?2??(x)b(x)dx)1/2x?=?(x)?1/2(C1?2??(x)b(x)dx)1/2

and solved implicitly:

C2+t??[?(x)(C1?2??(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

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