52.1.19 problem 17

Internal problem ID [8236]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 17
Date solved : Monday, January 27, 2025 at 03:46:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-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: 49

Order:=8; 
dsolve((x^2+2)*diff(y(x),x$2)+3*x*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}-\frac {17}{720} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 56

AsymptoticDSolveValue[(x^2+2)*D[y[x],{x,2}]+3*x*D[y[x],x]-y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (-\frac {17 x^7}{720}+\frac {7 x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {161 x^6}{5760}-\frac {7 x^4}{96}+\frac {x^2}{4}+1\right ) \]