46.2.21 problem 21

Internal problem ID [9558]
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 : Tuesday, September 30, 2025 at 06:20:43 PM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 52

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

\[ y = c_1 \,x^{{5}/{2}} \left (1+\frac {4}{7} x +\frac {4}{21} x^{2}+\frac {32}{693} x^{3}+\frac {80}{9009} x^{4}+\frac {64}{45045} x^{5}+\frac {64}{328185} x^{6}+\frac {1024}{43648605} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+\frac {1}{3} x -\frac {1}{6} x^{2}-\frac {1}{6} x^{3}-\frac {5}{72} x^{4}-\frac {7}{360} x^{5}-\frac {1}{240} x^{6}-\frac {11}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 113

ode=2*x*D[y[x],{x,2}]-(3+2*x)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.350 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) - (2*x + 3)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {11 x^{7}}{15120} - \frac {x^{6}}{240} - \frac {7 x^{5}}{360} - \frac {5 x^{4}}{72} - \frac {x^{3}}{6} - \frac {x^{2}}{6} + \frac {x}{3} + 1\right ) + C_{1} x^{\frac {5}{2}} \left (\frac {80 x^{4}}{9009} + \frac {32 x^{3}}{693} + \frac {4 x^{2}}{21} + \frac {4 x}{7} + 1\right ) + O\left (x^{8}\right ) \]