54.4.31 problem 1487

Internal problem ID [12753]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1487
Date solved : Wednesday, October 01, 2025 at 02:21:36 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} \left (2 x -1\right ) y^{\prime \prime \prime }+\left (x +4\right ) y^{\prime \prime }+2 y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

ode:=(2*x-1)*diff(diff(diff(y(x),x),x),x)+(x+4)*diff(diff(y(x),x),x)+2*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_3 +\int \frac {\left (2 c_1 x +c_2 \right ) {\mathrm e}^{\frac {x}{2}}}{\left (2 x -1\right )^{{3}/{4}}}d x \right ) {\mathrm e}^{-\frac {x}{2}}}{\left (2 x -1\right )^{{1}/{4}}} \]

✓ Mathematica. Time used: 60.288 (sec). Leaf size: 66

ode=2*D[y[x],x] + (4 + x)*D[y[x],{x,2}] + (-1 + 2*x)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^xe^{-\frac {K[1]}{2}} \left (\frac {2 \sqrt {2} c_1 K[1]}{(2 K[1]-1)^{5/4}}+c_2 L_{-\frac {1}{4}}^{\frac {5}{4}}\left (\frac {K[1]}{2}-\frac {1}{4}\right )\right )dK[1]+c_3 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + 4)*Derivative(y(x), (x, 2)) + (2*x - 1)*Derivative(y(x), (x, 3)) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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