69.22.17 problem 722

Internal problem ID [18483]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 17. Boundary value problems. Exercises page 163
Problem number : 722
Date solved : Thursday, October 02, 2025 at 03:14:10 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} 2 y^{\prime \prime }+4 x y^{\prime \prime \prime }+x^{2} y^{\prime \prime \prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 22

ode:=x^2*diff(diff(diff(diff(y(x),x),x),x),x)+4*x*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x) = 0; 
ic:=[y(1) = 0, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (-c_3 +\left (x -1\right ) c_4 \right ) \ln \left (x \right )+c_3 \left (x -1\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to (c_1-c_2) (x-1)+(c_2 x-c_1) \log (x) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{3} x \log {\left (x \right )} + C_{3} + C_{4} \log {\left (x \right )} + C_{4} + x \left (- C_{3} - C_{4}\right ) \]