A truly challenging ODE problem: find a proper integrating factor

[AdrianZ]

Find an integrating factor for the ODE y' + siny + xcosy + x = 0.
Any ideas?

 

[tkhunny]

Integrating Factor requires y' + yP(x) = Q(x)

Perhaps you should consider some other method.

 

[AdrianZ]

Integrating Factor requires y' + yP(x) = Q(x)

Perhaps you should consider some other method.

Not really. 1st order linear ODE's (which generally are of the form you mentioned) always have an integrating factor that can be found easily. the integrating factor in such a case would be a single variable function like F(x) that can be found from this formula: F(x)=exp(int(P(x))). But you can also find integrating factors for any first order ODE. any 1st order ODE which is generally of the form P(x,y)dx+Q(x,y)dy=0 has an integrating factor that can be found by solving a PDE iff It has a solution.

 

[galactus]

It is rather tricky. It requires the use of some trig identities.

\(\displaystyle \frac{dy}{dx}+sin(y)=-x(1+cos(y))\)

Note that \(\displaystyle tan(y/2)=\frac{sin(y)}{1+cos(y)}\) and \(\displaystyle \int \frac{1}{1+cos(y)}dy=tan(y/2)\)

The integrating factor is \(\displaystyle e^{\int dx}=e^{x}\)

You should end up with

\(\displaystyle \frac{d}{dx}[e^{x}tan(y/2)]=-xe^{x}\)

Integrate:

\(\displaystyle e^{x}tan(y/2)=(1-x)e^{x}+C\)

\(\displaystyle tan(y/2)=1-x+Ce^{-x}\)

\(\displaystyle y=2tan^{-1}\left(1-x+Ce^{-x}\right)\)

 

[tkhunny]

Not really. 1st order linear ODE's (which generally are of the form you mentioned) always have an integrating factor that can be found easily. the integrating factor in such a case would be a single variable function like F(x) that can be found from this formula: F(x)=exp(int(P(x))). But you can also find integrating factors for any first order ODE. any 1st order ODE which is generally of the form P(x,y)dx+Q(x,y)dy=0 has an integrating factor that can be found by solving a PDE iff It has a solution.

Excellent. You win the calling-me-out award. I do occasionally just make stuff up.