First order non-linear ODE

Hi everyone,

x^{2}y'-20+x^{2}y^{2}=0; y(1)=1

The question asks me to use substitution

**y=1/u*du/dx** to solve

So far I have

y'=-1/u^{2}*du/dx+1/u*d^{2}u/dx^{2}

problem is when I sub in for the x^{2}y^{2} term I'm going to get a derivative squared --> y^{2}=(1/u*du/dx)^{2}. Don't know how to resolve this. Please help.

Re: First order non-linear ODE

Re: First order non-linear ODE

Quote:

Originally Posted by

**lillybeans** Hi everyone,

x^{2}y'-20+x^{2}y^{2}=0; y(1)=1

The question asks me to use substitution

**y=1/u*du/dx** to solve

So far I have

y'=-1/u^{2}*du/dx+1/u*d^{2}u/dx^{2}

problem is when I sub in for the x^{2}y^{2} term I'm going to get a derivative squared --> y^{2}=(1/u*du/dx)^{2}. Don't know how to resolve this. Please help.

After simplification, the terme (1/u*du/dx)² disapears

Re: First order non-linear ODE

Thank you both I solved this problem BUT I have another one which I am really stuck on this time, supposedly a much easier problem.

There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.

1. y'''-9y''+14y'=x^{2}

2. y''-9y'+14y=x^{2}e^{4x}

For the first one I wrote:

yp=Px^{3}+Qx^{2}+Rx

Second one:

yp=(Px^{2}+Qx+R)e^{4x}

Neither of the answers are correct, according to the computer. Where did I go wrong?

Re: First order non-linear ODE

Re: First order non-linear ODE

Thank you very much for you help, but unfortunately that didn't work. I even checked my answer using Wolfram and it seems to be correct, maybe something is wrong with web assignment.

I used the method of undetermined coefficients to find the particular solution.

Re: First order non-linear ODE

Quote:

Originally Posted by

**lillybeans** Thank you both I solved this problem BUT I have another one which I am really stuck on this time, supposedly a much easier problem.

There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.

1. y'''-9y''+14y'=x^{2}

2. y''-9y'+14y=x^{2}e^{4x}

For the first one I wrote:

yp=Px^{3}+Qx^{2}+Rx

Second one:

yp=(Px^{2}+Qx+R)e^{4x}

Neither of the answers are correct, according to the computer. Where did I go wrong?

For Question 1, rewrite your equation as by letting . Then to solve for , the homogeneous solution has complementary function

So the homogeneous solution is .

Now as for the nonhomogeneous solution, use , which means and . Substituting into the DE gives

Therefore the nonhomogeneous solution is , so putting the homogeneous and nonhomogeneous solutions together gives the solution for as