74.16.13 problem 13

Internal problem ID [16519]
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 : 13
Date solved : Tuesday, January 28, 2025 at 09:10:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (2 x^{2}+2\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: 0.002 (sec). Leaf size: 39

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

\[ y = \left (1+\frac {3}{4} x^{2}-\frac {5}{32} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{3}-\frac {1}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to c_2 \left (-\frac {x^5}{32}+\frac {x^3}{12}+x\right )+c_1 \left (-\frac {5 x^4}{32}+\frac {3 x^2}{4}+1\right ) \]