40.1.11 problem 17

Internal problem ID [8587]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.1. page 174
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:39:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = 1+x -x^{2}-\frac {5}{6} x^{3}+\frac {2}{3} x^{4}+\frac {11}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to \frac {11 x^5}{24}+\frac {2 x^4}{3}-\frac {5 x^3}{6}-x^2+x+1 \]

✓ Sympy. Time used: 0.201 (sec). Leaf size: 31

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

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