46.2.18 problem 20

Internal problem ID [7321]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 20
Date solved : Monday, January 27, 2025 at 02:50:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 42

Order:=6; 
dsolve(3*t*(1+t)*diff(y(t),t$2)+t*diff(y(t),t)-y(t)=0,y(t),type='series',t=0);
 

\[ y = c_{1} t \left (1+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {1}{3} t +\operatorname {O}\left (t^{6}\right )\right ) \ln \left (t \right ) c_{2} +\left (1-\frac {1}{3} t -\frac {2}{9} t^{2}+\frac {7}{81} t^{3}-\frac {35}{729} t^{4}+\frac {91}{2916} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 43

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

\[ y(t)\to c_1 \left (\frac {1}{729} \left (-35 t^4+63 t^3-162 t^2+243 t+729\right )+\frac {1}{3} t \log (t)\right )+c_2 t \]