# Trouble in the second order ordinary differential equations with second oder coupled iterm

I ran into a system of ordinary differential equations of second order and was puzzled by it for a few days. It was originally easy to solve, but because the coupling term is second order, it cannot be solved directly by Mma. Here's the code I tried

eqs = {q[1]''[t] + q[1][t] -Sin[Pi t] (9.81 - q[1]''[t] Sin[ Pi t] - q[2]''[t] Sin[ 2*Pi t]) == 0 ,
q[2]''[t] + q[2][t] - Sin[2*Pi t] (9.81 - q[1]''[t] Sin[ Pi t] - q[2]''[t] Sin[ 2*Pi t]) == 0}
int = {q[1][0] == 0, q[2][0] == 0, q[1]'[0] == 0, q[2]'[0] == 0};
var = {q[1], q[2]};
exp = eqs~Join~int;
NDSolve[exp, var, {t, 0, 1}]


Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.

Infinity::indet: Indeterminate expression 0. ComplexInfinity ComplexInfinity encountered.

General::stop: Further output of Infinity::indet will be suppressed during this calculation.

Infinity::indet: Indeterminate expression 0. ComplexInfinity ComplexInfinity encountered.

NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.. I'm not sure if this system can be solved with mma, and I don't know how to solve it. I really hope you can help me.

It's because when transforming the system to the standard form required by ODE solver of NDSolve, a removable singularity is generated at $$t=0$$. (For more information check this post. ) There're at least 3 solutions for this problem. The easiest one is add SolveDelayed -> True:

NDSolveValue[exp, var, {t, 0, 1}, SolveDelayed -> True]


Or use a small number as starting point instead of $$0$$:

eps = 10^-3;
intappro = {q[1][eps] == 0, q[2][eps] == 0, q[1]'[eps] == 0, q[2]'[eps] == 0};
NDSolveValue[{eqs, intappro}, var, {t, eps, 1}]


Or use the asymptotic solution near $$t=0$$ as the initial condition:

asysol = AsymptoticDSolveValue[exp, var, {t, 0, 5}]

asyexpr = {q[1]@t, q[2]@t} == asysol // Thread

newic = {asyexpr, D[asyexpr, t]} /. t -> eps
sol = NDSolveValue[{eqs, newic}, var, {t, eps, 1}]

sol // ListLinePlot
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• I almost thought this equation was impossible to solve, and I really wanted to solve it myself. It seems that I am not familiar with the NDSolve solver and the process of solving differential equations. Thank you very much for your help and make me saw this problem clearly. Nov 21, 2019 at 8:37