87.26.16 problem 25

Internal problem ID [23850]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 25
Date solved : Thursday, October 02, 2025 at 09:45:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (x^{3}+x^{2}+x \right ) y^{\prime }+\left (1+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 46

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(x^3+x^2+x)*diff(y(x),x)+(1+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-3 x +\frac {7}{4} x^{2}-\frac {31}{36} x^{3}+\frac {21}{64} x^{4}-\frac {359}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (7 x -\frac {23}{4} x^{2}+\frac {71}{27} x^{3}-\frac {1367}{1152} x^{4}+\frac {34717}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 116

ode=x^2*D[y[x],{x,2}]-(x^3+x^2+x)*D[y[x],x]+(4*x+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (-\frac {359 x^5}{2880}+\frac {21 x^4}{64}-\frac {31 x^3}{36}+\frac {7 x^2}{4}-3 x+1\right )+c_2 \left (x \left (\frac {34717 x^5}{86400}-\frac {1367 x^4}{1152}+\frac {71 x^3}{27}-\frac {23 x^2}{4}+7 x\right )+x \left (-\frac {359 x^5}{2880}+\frac {21 x^4}{64}-\frac {31 x^3}{36}+\frac {7 x^2}{4}-3 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.412 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (4*x + 1)*y(x) - (x**3 + x**2 + x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {21 x^{4}}{64} - \frac {31 x^{3}}{36} + \frac {7 x^{2}}{4} - 3 x + 1\right ) + O\left (x^{6}\right ) \]