13.14.26 problem 27

Internal problem ID [2466]
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 : 27
Date solved : Monday, January 27, 2025 at 05:53:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \sin \left (t \right ) y^{\prime \prime }+\left (1-t \right ) y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.236 (sec). Leaf size: 44

Order:=6; 
dsolve(2*sin(t)*diff(y(t),t$2)+(1-t)*diff(y(t),t)-2*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \sqrt {t}\, \left (1+\frac {5}{6} t +\frac {17}{60} t^{2}+\frac {89}{1260} t^{3}+\frac {941}{45360} t^{4}+\frac {14989}{2494800} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+2 t +t^{2}+\frac {4}{15} t^{3}+\frac {1}{14} t^{4}+\frac {101}{4725} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 1303

AsymptoticDSolveValue[2*sin(t)*D[y[t],{t,2}]+(1-t)*D[y[t],t]-2*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to \left (\frac {\left (\frac {2 \sin -1}{4 \sin ^2}+\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+3}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+4}{2 \sin }-\frac {1}{\sin }\right ) t^5}{\left (\frac {(2 \sin -1) \left (\frac {2 \sin -1}{2 \sin }+1\right )}{2 \sin }+\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+1\right ) \left (\frac {2 \sin -1}{2 \sin }+2\right )+\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+2\right ) \left (\frac {2 \sin -1}{2 \sin }+3\right )+\frac {\frac {2 \sin -1}{2 \sin }+3}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+3\right ) \left (\frac {2 \sin -1}{2 \sin }+4\right )+\frac {\frac {2 \sin -1}{2 \sin }+4}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+4\right ) \left (\frac {2 \sin -1}{2 \sin }+5\right )+\frac {\frac {2 \sin -1}{2 \sin }+5}{2 \sin }\right )}-\frac {\left (\frac {2 \sin -1}{4 \sin ^2}+\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+3}{2 \sin }-\frac {1}{\sin }\right ) t^4}{\left (\frac {(2 \sin -1) \left (\frac {2 \sin -1}{2 \sin }+1\right )}{2 \sin }+\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+1\right ) \left (\frac {2 \sin -1}{2 \sin }+2\right )+\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+2\right ) \left (\frac {2 \sin -1}{2 \sin }+3\right )+\frac {\frac {2 \sin -1}{2 \sin }+3}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+3\right ) \left (\frac {2 \sin -1}{2 \sin }+4\right )+\frac {\frac {2 \sin -1}{2 \sin }+4}{2 \sin }\right )}+\frac {\left (\frac {2 \sin -1}{4 \sin ^2}+\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }-\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }-\frac {1}{\sin }\right ) t^3}{\left (\frac {(2 \sin -1) \left (\frac {2 \sin -1}{2 \sin }+1\right )}{2 \sin }+\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+1\right ) \left (\frac {2 \sin -1}{2 \sin }+2\right )+\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+2\right ) \left (\frac {2 \sin -1}{2 \sin }+3\right )+\frac {\frac {2 \sin -1}{2 \sin }+3}{2 \sin }\right )}-\frac {\left (\frac {2 \sin -1}{4 \sin ^2}+\frac {1}{\sin }\right ) \left (-\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }-\frac {1}{\sin }\right ) t^2}{\left (\frac {(2 \sin -1) \left (\frac {2 \sin -1}{2 \sin }+1\right )}{2 \sin }+\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }\right ) \left (\left (\frac {2 \sin -1}{2 \sin }+1\right ) \left (\frac {2 \sin -1}{2 \sin }+2\right )+\frac {\frac {2 \sin -1}{2 \sin }+2}{2 \sin }\right )}+\frac {\left (\frac {2 \sin -1}{4 \sin ^2}+\frac {1}{\sin }\right ) t}{\frac {(2 \sin -1) \left (\frac {2 \sin -1}{2 \sin }+1\right )}{2 \sin }+\frac {\frac {2 \sin -1}{2 \sin }+1}{2 \sin }}+1\right ) c_1 t^{\frac {2 \sin -1}{2 \sin }}+\left (\frac {45 t^5}{\left (2+\frac {1}{\sin }\right ) \left (6+\frac {3}{2 \sin }\right ) \left (12+\frac {2}{\sin }\right ) \left (20+\frac {5}{2 \sin }\right ) \sin ^4}+\frac {15 t^4}{\left (2+\frac {1}{\sin }\right ) \left (6+\frac {3}{2 \sin }\right ) \left (12+\frac {2}{\sin }\right ) \sin ^3}+\frac {6 t^3}{\left (2+\frac {1}{\sin }\right ) \left (6+\frac {3}{2 \sin }\right ) \sin ^2}+\frac {3 t^2}{\left (2+\frac {1}{\sin }\right ) \sin }+2 t+1\right ) c_2 \]