23.1.182 problem 182

Internal problem ID [4789]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 182
Date solved : Tuesday, September 30, 2025 at 08:36:13 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+a_{0} +a_{1} x +\left (a_{2} +a_{3} x y\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 403

ode:=x*diff(y(x),x)+a__0+a__1*x+(a__2+a__3*x*y(x))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {4 \left (a_{1}^{3} a_{3} \left (a_{3} a_{0} -a_{2} \sqrt {-a_{1} a_{3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\frac {c_1 \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )}{4}+a_{1}^{3} a_{3} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_1 \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right )}{2}\right ) a_{1}}{4 a_{1}^{3} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\sqrt {-a_{1} a_{3}}\, c_1 \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+2 \left (-2 a_{1}^{2} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_1 \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right )\right ) a_{1}} \]

✓ Mathematica. Time used: 0.252 (sec). Leaf size: 421

ode=x*D[y[x],x]+a0+a1*x+(a2+a3*x*y[x])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \left (\sqrt {\text {a1}} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )}\\ y(x)&\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(a0 + a1*x + x*Derivative(y(x), x) + (a2 + a3*x*y(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out