52.2.21 problem 21

Internal problem ID [8272]
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 : 21
Date solved : Monday, January 27, 2025 at 03:47:35 PM
CAS classification : [_Laguerre]

\begin{align*} 2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 52

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

\[ y = c_{1} x^{{5}/{2}} \left (1+\frac {4}{7} x +\frac {4}{21} x^{2}+\frac {32}{693} x^{3}+\frac {80}{9009} x^{4}+\frac {64}{45045} x^{5}+\frac {64}{328185} x^{6}+\frac {1024}{43648605} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{3} x -\frac {1}{6} x^{2}-\frac {1}{6} x^{3}-\frac {5}{72} x^{4}-\frac {7}{360} x^{5}-\frac {1}{240} x^{6}-\frac {11}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 113

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

\[ y(x)\to c_2 \left (-\frac {11 x^7}{15120}-\frac {x^6}{240}-\frac {7 x^5}{360}-\frac {5 x^4}{72}-\frac {x^3}{6}-\frac {x^2}{6}+\frac {x}{3}+1\right )+c_1 \left (\frac {1024 x^7}{43648605}+\frac {64 x^6}{328185}+\frac {64 x^5}{45045}+\frac {80 x^4}{9009}+\frac {32 x^3}{693}+\frac {4 x^2}{21}+\frac {4 x}{7}+1\right ) x^{5/2} \]