12.13.33 problem 33

Internal problem ID [1924]
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 : 33
Date solved : Saturday, March 29, 2025 at 11:43:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\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 )&=1\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

Order:=6; 
ode:=diff(diff(y(x),x),x)-3*x*diff(y(x),x)+(2*x^2+5)*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = -2; 
dsolve([ode,ic],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to \frac {x^5}{3}-\frac {3 x^4}{8}+\frac {2 x^3}{3}-\frac {5 x^2}{2}-2 x+1 \]

✓ Sympy. Time used: 0.795 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + (2*x**2 + 5)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {x^{5} r{\left (3 \right )}}{5} + C_{2} \left (- \frac {3 x^{4}}{8} - \frac {5 x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{10}\right ) + O\left (x^{6}\right ) \]