4.14.3 problem 3

Internal problem ID [1385]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 04:33:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 16

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+3*ln(x)*y(x) = 0; 
ic:=[y(1) = 2, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = 2-\left (-1+x \right )^{3}+\frac {7}{4} \left (-1+x \right )^{4}-\frac {49}{20} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 30

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

\[ y(x)\to -\frac {49}{20} (x-1)^5+\frac {7}{4} (x-1)^4-(x-1)^3+2 \]

✓ Sympy. Time used: 0.374 (sec). Leaf size: 112

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

\[ y{\left (x \right )} = C_{2} \left (x + \left (x - 1\right )^{4} \log {\left (x + 1 \right )} - \frac {17 \left (x - 1\right )^{4}}{12} - \frac {\left (x - 1\right )^{3} \log {\left (x + 1 \right )}}{2} + \frac {7 \left (x - 1\right )^{3}}{6} - \left (x - 1\right )^{2} - 1\right ) + C_{1} \left (\frac {3 \left (x - 1\right )^{4} \log {\left (x + 1 \right )}^{2}}{8} - \frac {5 \left (x - 1\right )^{4} \log {\left (x + 1 \right )}}{2} + 2 \left (x - 1\right )^{3} \log {\left (x + 1 \right )} - \frac {3 \left (x - 1\right )^{2} \log {\left (x + 1 \right )}}{2} + 1\right ) + O\left (x^{6}\right ) \]