12.11.5 problem 15

Internal problem ID [1844]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 15
Date solved : Monday, January 27, 2025 at 05:37:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 37

Order:=6; 
dsolve((1+3*x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (x^{4}-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

AsymptoticDSolveValue[(1+3*x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {4 x^5}{15}-\frac {x^3}{3}+x\right )+c_1 \left (x^4-2 x^2+1\right ) \]