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 : Monday, January 27, 2025 at 05:53:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.186 (sec). Leaf size: 228

Order:=6; 
dsolve(t^3*diff(y(t),t$2)+sin(t^3)*diff(y(t),t)+t*y(t)=0,y(t),type='series',t=0);
 

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[t^3*D[y[t],{t,2}]+Sin[t^3]*D[y[t],t]+t*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to \left (\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right ) t^5}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right ) \left (1+\left (4-(-1)^{2/3}\right ) \left (5-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right ) t^4}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right ) \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right ) t^3}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right ) \left (1+\left (2-(-1)^{2/3}\right ) \left (3-(-1)^{2/3}\right )\right )}-\frac {(-1)^{2/3} \left (1-(-1)^{2/3}\right ) t^2}{\left (1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )\right ) \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+\frac {(-1)^{2/3} t}{1-(-1)^{2/3} \left (1-(-1)^{2/3}\right )}+1\right ) c_1 t^{-(-1)^{2/3}}+\left (-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right ) t^5}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right ) \left (1+\left (4+\sqrt [3]{-1}\right ) \left (5+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right ) t^4}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right ) \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right ) t^3}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right ) \left (1+\left (2+\sqrt [3]{-1}\right ) \left (3+\sqrt [3]{-1}\right )\right )}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) t^2}{\left (1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {\sqrt [3]{-1} t}{1+\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )}+1\right ) c_2 t^{\sqrt [3]{-1}} \]