6.13.35 problem 35

Internal problem ID [1926]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 05:21:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=diff(diff(y(x),x),x)-2*x*diff(y(x),x)-(3*x^2+2)*y(x) = 0; 
ic:=[y(0) = 2, D(y)(0) = -5]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 2-5 x +2 x^{2}-\frac {10}{3} x^{3}+\frac {3}{2} x^{4}-\frac {25}{12} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {25 x^5}{12}+\frac {3 x^4}{2}-\frac {10 x^3}{3}+2 x^2-5 x+2 \]

✓ Sympy. Time used: 0.260 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) - (3*x**2 + 2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): -5} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {2 x^{5} r{\left (3 \right )}}{5} + C_{2} \left (\frac {3 x^{4}}{4} + x^{2} + 1\right ) + C_{1} x \left (\frac {3 x^{4}}{20} + 1\right ) + O\left (x^{6}\right ) \]