56.4.41 problem 38

Internal problem ID [8930]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 38
Date solved : Sunday, March 30, 2025 at 01:55:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

Order:=6; 
ode:=2*x^2*(x+2)*diff(diff(y(x),x),x)+5*x^2*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{4} x +\frac {15}{32} x^{2}-\frac {35}{128} x^{3}+\frac {315}{2048} x^{4}-\frac {693}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x -\frac {13}{64} x^{2}+\frac {101}{768} x^{3}-\frac {641}{8192} x^{4}+\frac {7303}{163840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 134

ode=2*x^2*(2+x)*D[y[x],{x,2}] +5*x^2*D[y[x],x]+(1+x)*y[x] == 0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {7303 x^5}{163840}-\frac {641 x^4}{8192}+\frac {101 x^3}{768}-\frac {13 x^2}{64}+\frac {x}{4}\right )+\sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.025 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 2)*Derivative(y(x), (x, 2)) + 5*x**2*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {21875 x^{4}}{3072} - \frac {625 x^{3}}{96} + \frac {75 x^{2}}{16} - \frac {5 x}{2} + 1\right ) + O\left (x^{6}\right ) \]