13.14.6 problem 6

Internal problem ID [2446]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:01:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{3} y^{\prime \prime }+\sin \left (t^{3}\right ) y^{\prime }+t y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.126 (sec). Leaf size: 228

Order:=6; 
ode:=t^3*diff(diff(y(t),t),t)+sin(t^3)*diff(y(t),t)+t*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \sqrt {t}\, \left (c_2 \,t^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} t +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} t^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} t^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} t^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_1 \,t^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} t +\frac {\sqrt {3}+3 i}{8 \sqrt {3}+16 i} t^{2}+\frac {-\sqrt {3}-5 i}{48 \sqrt {3}+96 i} t^{3}+\frac {3 i \sqrt {3}-8}{576 i \sqrt {3}-480} t^{4}-\frac {1}{3840} \frac {\left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+9 i\right )}{\left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+2 i\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 886

ode=t^3*D[y[t],{t,2}]+Sin[t^3]*D[y[t],t]+t*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**3*Derivative(y(t), (t, 2)) + t*y(t) + sin(t**3)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE t**3*Derivative(y(t), (t, 2)) + t*y(t) + sin(t**3)*Derivative(y(t), t) does not match hint 2nd_power_series_regular