62.36.8 problem Ex 8

Internal problem ID [12935]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 8
Date solved : Monday, March 31, 2025 at 07:27:27 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 22

ode:=x^2*diff(diff(diff(y(x),x),x),x)-5*x*diff(diff(y(x),x),x)+(4*x^4+5)*diff(y(x),x)-8*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{2}+c_2 \cos \left (x^{2}\right )+c_3 \sin \left (x^{2}\right ) \]

✓ Mathematica. Time used: 0.493 (sec). Leaf size: 98

ode=x^2*D[y[x],{x,3}]-5*x*D[y[x],{x,2}]+(4*x^4+5)*D[y[x],x]-8*x^3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_3 x^2 \int _1^x\frac {e^{-i K[2]^2} \left (K[2]^2-i\right ) \int _1^{K[2]}\frac {e^{2 i K[1]^2} K[1]^5}{\left (K[1]^2-i\right )^2}dK[1]}{K[2]^3}dK[2]+c_1 x^2+\frac {1}{2} i c_2 e^{-i x^2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**3*y(x) + x**2*Derivative(y(x), (x, 3)) - 5*x*Derivative(y(x), (x, 2)) + (4*x**4 + 5)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(8*x**2*y(x) - x*Derivative(y(x), (x, 3)) + 5*Derivative(y(x), (x, 2)))/(4*x**4 + 5) + Derivative(y(x), x) cannot be solved by the factorable group method