Hunt for exact solutions of second order ordinary differential equations with varying coefficients.

Solution 1:

I think there also be some more examples:

$1.$ ODE of the form $\dfrac{d^2y}{dx^2}+(a_4x^4+a_3x^3+a_2x^2+a_1x+a_0)y=0$ , $a_4\neq0$ can first convert to $\dfrac{d^2y}{dt^2}+(b_4t^4+b_2t^2+b_1t+b_0)y=0$ and then relates to Heun's Triconfluent Equation as above. The case of $a_4=0$ and $a_3\neq0$ is a big headache.

$2.$ ODE of the form $(x+a)^2(x+b)^2\dfrac{d^2y}{dx^2}+(c_3x^3+c_2x^2+c_1x+c_0)y=0$ , $c_3\neq0$ can convert to Heun's Confluent Equation by letting $y=(x+a)^p(x+b)^qu$ with choosing suitable values of $p$ and $q$ similar to Differential equation with nasty coefficients $ x^2(1-x)^2 y'' + (Ax + b)y = 0 $.

Anyway, I think the most difficulties appear in e.g. "slipped fingers from Heun-type ODEs" , i.e. for example in https://math.stackexchange.com/questions/2944492, Does Heun's differential equation have other known types confluent approach?, an odd question about solving ODE by MATLAB, Solutions in terms of the hypergeometric functions, differential equation nondevelopable, Solving differential equation, Why can't I solve this homogenous second order differential equation?, Special Differential Equation, solving second order differential equation, Solve the given initial value problem.I need your help., differential equation - solving a second-order ODE with variable coefficients, etc. Welcome to challenge! Good luck!