74.16.16 problem 16

Internal problem ID [16522]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 09:10:09 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*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {x^5}{4}+\frac {5 x^4}{4}-\frac {x^3}{3}+3 x^2+2 x-2 \]