highP6 = D[y[t], {t, 3}] - Power[(Cos[t]), 3]*y[t] == 0;
ic1 = y[0] == a;
ic2 = y'[0] == -a;
ic3 = y''[0] == 0;
DSolve[{highP6, ic1, ic2, ic3}, y[t], t]
I cannot get the solution using the above code? Any help!
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Sign up to join this communityhighP6 = D[y[t], {t, 3}] - Power[(Cos[t]), 3]*y[t] == 0;
ic1 = y[0] == a;
ic2 = y'[0] == -a;
ic3 = y''[0] == 0;
DSolve[{highP6, ic1, ic2, ic3}, y[t], t]
I cannot get the solution using the above code? Any help!
Since this is variable coefficient ODE, you can try series solution. The analytical series solution will be accurate "near" the point of expansion, which has to be where the initial conditions are. More terms makes it more accurate.
ClearAll[y,t,a];
nTerms=10;
pt=0;
ic={y[0]->a,y'[0]->-a,y''[0]->0};
ode=y'''[t]- Cos[t]^3 y[t];
s0=Series[ode,{t,pt,nTerms}];
s0=s0/.ic;
roots=Solve@LogicalExpand[s0==0];
s1=Series[y[t],{t,pt,nTerms+2}];
sol=Normal[s1]/.ic/.roots[[1]]
Compare to Numerical
a=1;
nSol=NDSolve[{ode==0,y[0]==a,y'[0]==-a,y''[0]==0},y,{t,0,2}];
Plot[{sol/.a->1,Evaluate[y[t]/.nSol]},{t,0,2},
PlotStyle->{Red,Blue},PlotLegends->{"Series","Numerical"}]
Reference: solving-a-second-order-non-linear-differential-equation
Comment
Like Mathematica, Maple is also unable to solve the ode.
restart;
ode:=diff(y(t),t$3)-cos(t)^3*y(t)=0;
dsolve(ode);
DESol is a data structure to represent the solution of a differential equation.