23.5.171 problem 171

Internal problem ID [6780]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 171
Date solved : Tuesday, September 30, 2025 at 03:51:38 PM
CAS classification : [[_high_order, _quadrature]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 48

ode:=(x+a)^2*diff(diff(diff(diff(y(x),x),x),x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\ln \left (a +x \right ) \left (a +x \right )^{2}}{2}+\frac {c_1 \,x^{3}}{6}+\frac {\left (3+2 c_2 \right ) x^{2}}{4}+\frac {\left (3 a +2 c_3 \right ) x}{2}+\frac {a^{2}}{4}+c_4 \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to \frac {1}{4} \left (x (2 a+3 x)-2 (a+x)^2 \log (a+x)\right )+c_4 x^3+c_3 x^2+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 0.314 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} x^{2} + C_{4} x^{3} - \frac {a^{2} \log {\left (a + x \right )}}{2} - \frac {a^{2}}{6} - a x \log {\left (a + x \right )} - \frac {x^{2} \log {\left (a + x \right )}}{2} + \frac {x^{2}}{4} \]