50.19.16 problem 4(c)

Internal problem ID [8124]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 4(c)
Date solved : Sunday, March 30, 2025 at 12:46:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+\left (x +1\right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 52

Order:=8; 
ode:=2*x*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1-\frac {7}{6} x +\frac {21}{40} x^{2}-\frac {11}{80} x^{3}+\frac {143}{5760} x^{4}-\frac {13}{3840} x^{5}+\frac {17}{46080} x^{6}-\frac {323}{9676800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-3 x +2 x^{2}-\frac {2}{3} x^{3}+\frac {1}{7} x^{4}-\frac {1}{45} x^{5}+\frac {4}{1485} x^{6}-\frac {4}{15015} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 106

ode=2*x*x*D[y[x],{x,2}]+(x+1)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {1386072 x^7}{35}+\frac {20088 x^6}{5}-\frac {2511 x^5}{5}+81 x^4-18 x^3+6 x^2-3 x+1\right )+c_2 e^{\left .\frac {1}{2}\right /x} \left (\frac {257243688 x^7}{35}+\frac {2381886 x^6}{5}+\frac {176436 x^5}{5}+3042 x^4+312 x^3+39 x^2+6 x+1\right ) x^{3/2} \]

✓ Sympy. Time used: 1.030 (sec). Leaf size: 95

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {4 x^{7}}{15015} + \frac {4 x^{6}}{1485} - \frac {x^{5}}{45} + \frac {x^{4}}{7} - \frac {2 x^{3}}{3} + 2 x^{2} - 3 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {17 x^{6}}{46080} - \frac {13 x^{5}}{3840} + \frac {143 x^{4}}{5760} - \frac {11 x^{3}}{80} + \frac {21 x^{2}}{40} - \frac {7 x}{6} + 1\right ) + O\left (x^{8}\right ) \]