52.2.25 problem 25

Internal problem ID [8276]
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. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 25
Date solved : Monday, January 27, 2025 at 03:47:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }-y x&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 36

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

\[ y = c_{1} \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 56

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

\[ y(x)\to c_1 \left (\frac {x^5}{720}+\frac {x^3}{24}+\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^6}{5040}+\frac {x^4}{120}+\frac {x^2}{6}+1\right ) \]