74.17.3 problem 3

Internal problem ID [16534]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 09:10:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-3 x -4\right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+\left (x^{2}-1\right ) 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: 59

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

\[ y = \left (1-\frac {1}{8} x^{2}+\frac {1}{24} x^{3}+\frac {1}{192} x^{4}-\frac {1}{640} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{8} x^{2}-\frac {1}{24} x^{3}+\frac {1}{48} x^{4}+\frac {1}{960} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (-\frac {x^5}{640}+\frac {x^4}{192}+\frac {x^3}{24}-\frac {x^2}{8}+1\right )+c_2 \left (\frac {x^5}{960}+\frac {x^4}{48}-\frac {x^3}{24}-\frac {x^2}{8}+x\right ) \]