Consider the following first order ODE
$$y'(t) = a\,y(t) – b\, y^{\frac{3}{2}}(t) + h(t).$$
where $a$ and $b$ are positive constants. $h$ is a $C^1$ function. I solved the homogenous part by methods applicable to Bernoulli-type equations and am now wondering how to proceed to the inhomogenous part. I tried variation of constants, which failed.
Is there a generic way to go on? I also tried to adapt the method that one can use for Riccati equations, setting $y = -\dfrac{u}{b u'}$ to arrive at another but homogenous equation, as it happens in de Riccati case.
Could someone suggest a method?
Best Answer
For nonlinear differential equations in general, solving the "homogeneous part" does not help at all with the "inhomogeneous part". There is no way to get from the general solution of the homogeneous equation and a particular solution of the inhomogeneous equation to a general solution of the inhomogeneous equation.
In this case, even for something as simple as $h(t)=t$ I don't believe there is a closed-form solution. Maple doesn't find one, nor any symmetries. This is a Chini differential equation, but the Chini invariant is not constant.