23.3.566 problem 574

Internal problem ID [6280]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 574
Date solved : Tuesday, September 30, 2025 at 02:42:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

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

\[ y = c_1 \sinh \left (\frac {1}{a -x}\right )+c_2 \cosh \left (\frac {1}{a -x}\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to c_1 \cosh \left (\frac {1}{a-x}\right )+i c_2 \sinh \left (\frac {1}{a-x}\right ) \end{align*}

✓ Sympy. Time used: 0.268 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = \frac {\frac {C_{1} \sqrt {\frac {i}{- a + x}} J_{- \frac {1}{2}}\left (\frac {i}{- a + x}\right )}{\sqrt {- \frac {i}{- a + x}}} + C_{2} Y_{- \frac {1}{2}}\left (- \frac {i}{- a + x}\right )}{\sqrt {- a + x}} \]