74.18.59 problem 65

Internal problem ID [16616]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 65
Date solved : Tuesday, January 28, 2025 at 09:12:53 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (2 x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-3 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: 39

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

\[ y = \left (1-\frac {3}{2} x^{2}-\frac {5}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (-\frac {x^5}{8}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {5 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]