56.4.47 problem 44

Internal problem ID [8936]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 44
Date solved : Sunday, March 30, 2025 at 01:55:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

Order:=6; 
ode:=cos(x)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 49

ode=Cos[x]*D[y[x],{x,2}]+2*x*D[y[x],x]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{40}+\frac {x^3}{6}+1\right )+c_2 \left (\frac {x^5}{20}+\frac {x^4}{12}-\frac {x^3}{3}+x\right ) \]

✓ Sympy. Time used: 1.031 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{20 \cos ^{2}{\left (x \right )}} + \frac {x^{3}}{6 \cos {\left (x \right )}} + 1\right ) + C_{1} x \left (\frac {x^{4}}{10 \cos ^{2}{\left (x \right )}} + \frac {x^{3}}{12 \cos {\left (x \right )}} - \frac {x^{2}}{3 \cos {\left (x \right )}} + 1\right ) + O\left (x^{6}\right ) \]