23.3.341 problem 344

Internal problem ID [6055]
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 : 344
Date solved : Tuesday, September 30, 2025 at 02:20:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y \left (\operatorname {a2} +\operatorname {b2} \,x^{k}+\operatorname {c2} \,x^{2 k}+\left (-1+\operatorname {a1} +\operatorname {b1} \,x^{k}\right ) f \left (x \right )+f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )+x \left (\operatorname {a1} +\operatorname {b1} \,x^{k}+2 f \left (x \right )\right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=y(x)*(a2+b2*x^k+c2*x^(2*k)+(-1+a1+b1*x^k)*f(x)+f(x)^2+diff(f(x),x))+x*(a1+b1*x^k+2*f(x))*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*(a2 + b2*x^k + c2*x^(2*k) + (-1 + a1 + b1*x^k)*f[x] + f[x]^2 + D[f[x],x]) + x*(a1 + b1*x^k + 2*f[x])*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b1 = symbols("b1") 
b2 = symbols("b2") 
c2 = symbols("c2") 
k = symbols("k") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a1 + b1*x**k + 2*f(x))*Derivative(y(x), x) + (a2 + b2*x**k + c2*x**(2*k) + (a1 + b1*x**k - 1)*f(x) + f(x)**2 + Derivative(f(x), x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a1*f(x)*y(x) - a2*y(x) - b1*x**k*f(x)*y(