10.14.4 problem 4

Internal problem ID [1386]
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 : 4
Date solved : Tuesday, March 04, 2025 at 12:34:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

Order:=6; 
ode:=diff(diff(y(x),x),x)+x^2*diff(y(x),x)+sin(x)*y(x) = 0; 
ic:=y(0) = a__0, D(y)(0) = a__1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = a_{0} +a_{1} x -\frac {1}{6} a_{0} x^{3}-\frac {1}{6} a_{1} x^{4}+\frac {1}{120} a_{0} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

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

ode=D[y[x],{x,2}]+x^2*D[y[x],x]+Sin[x]*y[x]==0; 
ic={y[0]==a0,Derivative[1][y][0] ==a1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {\text {a0} x^5}{120}-\frac {\text {a0} x^3}{6}+\text {a0}-\frac {\text {a1} x^4}{6}+\text {a1} x \]

✓ Sympy. Time used: 1.702 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + y(x)*sin(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): a__0, Subs(Derivative(y(x), x), x, 0): a__1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} \sin ^{2}{\left (x \right )}}{24} - \frac {x^{2} \sin {\left (x \right )}}{2} + 1\right ) + C_{1} x \left (- \frac {x^{3}}{12} - \frac {x^{2} \sin {\left (x \right )}}{6} + 1\right ) + O\left (x^{6}\right ) \]