ode internal name "exact_nonlinear_second_order_ode_with_integrating_factor"
3.4.2.2.1 Introduction Not implemented yet. The above section showed how to solve the ode \(F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =0\) once it is determined it is exact as is, which is by finding the first integral \(R\). But the real problem is what to do if the ode is not exact as is?. Given the second order nonlinear ode\[ F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =0 \] Which is not exact as is (using the earlier test shown), then we need to either find an integrating factor \(\mu \) to make it exact (this integrating factor might or might not exist) or try to find the first integral directly without finding an integrating factor first. There are few papers that show how to do this for some types of nonlinear second order odes.
Using an integrating factor approach, If we are able to find \(\mu \), then the ode can now be solved as type "second order integrable as is" or as type "exact nonlinear second order ode" as shown in the above section. (need to merge these types).
As mentioned earlier, an ode \(F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =0\) is called exact if there exists a function \(R\left ( x,y,y^{\prime }\right ) \) (called first integral) with order one less than the order of the ode, such that \[ F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =\frac {d}{dx}R\left ( x,y,y^{\prime }\right ) \] If the ode is not exact, then we need to find an integrating factor of any of these forms \(\mu \left ( x\right ) ,\mu \left ( y\right ) ,\mu \left ( y^{\prime }\right ) ,\mu \left ( x,y\right ) ,\mu \left ( x,y^{\prime }\right ) ,\mu \left ( y,y^{\prime }\right ) \) such that \(\mu F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) \) is now exact and hence \[ \mu F\left ( x,y,y^{\prime },y^{\prime \prime }\right ) =\frac {d}{dx}R\left ( x,y,y^{\prime }\right ) \] The main difficulty is how to find \(\mu \). Few papers were written on this (but I found them all not very clear as they give no examples).
Finding \(\mu \) with first order ODE is easy. But not so easy with second order ode’s. Note that in the above, an integrating factor of the form \(\mu =\mu \left ( x,y,y^{\prime }\right ) \) will not be considered as finding such an integrating factor requires solving a pde which is harder than solving the original ode. There two relations are important in order to find \(\mu \)\begin {align} R & =G\left ( x,y\right ) +\int \mu dy^{\prime }\tag {1}\\ & =G\left ( x,y\right ) +\int \mu dp\nonumber \end {align}
Where \(p=y^{\prime }\) and \(G\) is some function to be determined. As was derived in the introduction of the earlier section, we also have the relation\begin {equation} R_{x}+y^{\prime }R_{y}+\Phi R_{y^{\prime }}=0 \tag {2} \end {equation}
3.4.2.2.2 Integrating factors by inspection. These are not yet implemented. Before going through the formal way to find \(\mu \) for non exact second order nonlinear ode, there is a table given by Murphy which we can utilize before searching for \(\mu \) as a lookup table. Writing the ode as \(y^{\prime \prime }+g\left ( x,y,y^{\prime }\right ) =0\) the table is
| \(g\left ( x,y,y^{\prime }\right ) \) form | integrating factor |
| \(g\left ( y\right ) \) (i.e. function of \(y\) only) | \(y^{\prime }\) |
| \(g\left ( y^{\prime }\right ) \) (i.e. function of \(y^{\prime }\) only) | \(\frac {y^{\prime }}{g}\) |
| \(p\left ( x,y\right ) y^{\prime }+Q\left ( x,y\right ) \left ( y^{\prime }\right ) ^{2}\) | \(\frac {1}{y^{\prime }}\) |
| \(p\left ( x,y\right ) +Q\left ( x,y\right ) y^{\prime }\) such that \(\frac {\partial p}{\partial y}=\frac {\partial Q}{\partial x}\) | \(\frac {1}{y^{\prime }}\) |
The above integrating factors are from Murphy book page 165.
3.4.2.2.3 Integrating factor \(\mu \left ( x\right ) \) that depends on \(x\) only Not implemented.
3.4.2.2.4 Integrating factor \(\mu \left ( y\right ) \) that depends on \(y\) only Not implemented.
3.4.2.2.5 Integrating factor \(\mu \left ( y^{\prime }\right ) \) that depends on \(y^{\prime }\) only Not implemented.
3.4.2.2.6 Integrating factor \(\mu \left ( x,y\right ) \) Not implemented.
3.4.2.2.7 Integrating factor \(\mu \left ( x,y^{\prime }\right ) \) Not implemented.
3.4.2.2.8 Integrating factor \(\mu \left ( y,y^{\prime }\right ) \) Not implemented.
3.4.2.2.9 Checking if an integrating factor exists (but not find it) An example is
\[ xy\left ( 2x+y\right ) y^{\prime \prime }+\left ( x^{2}+xy\right ) y^{\prime }+\left ( 3xy+y^{2}\right ) =0 \] to do.