56.4.16 problem 16

Internal problem ID [8905]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 16
Date solved : Sunday, March 30, 2025 at 01:52:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 54

Order:=6; 
ode:=(1+x)*(3*x-1)*diff(diff(y(x),x),x)+cos(x)*diff(y(x),x)-3*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {53 x^5}{40}-\frac {5 x^4}{8}-\frac {x^3}{2}+1\right )+c_2 \left (\frac {7 x^5}{6}+\frac {7 x^4}{12}+\frac {x^3}{2}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 1.449 (sec). Leaf size: 94

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4} \cos {\left (x \right )}}{8} - \frac {x^{4}}{2} - \frac {x^{3}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3} \cos ^{3}{\left (x \right )}}{24} + \frac {x^{3} \cos ^{2}{\left (x \right )}}{4} + \frac {7 x^{3} \cos {\left (x \right )}}{12} - \frac {x^{3}}{4} + \frac {x^{2} \cos ^{2}{\left (x \right )}}{6} + \frac {x^{2} \cos {\left (x \right )}}{3} + \frac {x \cos {\left (x \right )}}{2} + 1\right ) + O\left (x^{6}\right ) \]