61.4.19 problem 40

Internal problem ID [12045]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and exponential functions
Problem number : 40
Date solved : Sunday, March 30, 2025 at 10:21:09 PM
CAS classification : [_Riccati]

\begin{align*} x^{4} \left (y^{\prime }-y^{2}\right )&=a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}} \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 280

ode:=x^4*(diff(y(x),x)-y(x)^2) = a+b*exp(k/x)+c*exp(2*k/x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (i \sqrt {a}+\frac {k}{2}\right ) \sqrt {c}-\frac {i b}{2}\right ) \operatorname {WhittakerM}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )-c_1 k \operatorname {WhittakerW}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) \sqrt {c}+\left (i {\mathrm e}^{\frac {k}{x}} c +\left (-\frac {k}{2}-x \right ) \sqrt {c}+\frac {i b}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )}{\sqrt {c}\, x^{2} \left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )} \]

✓ Mathematica. Time used: 2.079 (sec). Leaf size: 940

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (a + b*exp(k/x) + c*exp(2*k/x) + x**4*y(x)**2)/x**4 cannot be solved by the factorable group method