15.4.3 problem 3

Internal problem ID [2916]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 3
Date solved : Sunday, March 30, 2025 at 12:55:13 AM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 85

ode:=a__1*x+b__1*y(x)+c__1+(b__1*x+b__2*y(x)+c__2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-c_1 a_{1} \left (b_{1} x +c_{2} \right ) b_{2} +c_1 \,b_{1}^{3} x +c_1 \,b_{1}^{2} c_{2} -\sqrt {-\left (a_{1} b_{2} -b_{1}^{2}\right ) \left (\left (a_{1} x +c_{1} \right ) b_{2} -b_{1}^{2} x -b_{1} c_{2} \right )^{2} c_1^{2}+b_{2}}}{b_{2} c_1 \left (a_{1} b_{2} -b_{1}^{2}\right )} \]

✓ Mathematica. Time used: 17.085 (sec). Leaf size: 106

ode=(a1*x+b1*y[x]+c1)+(b1*x+b2*y[x]+c2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\frac {\sqrt {-x (\text {a1} x+2 \text {c1})+\frac {(\text {b1} x+\text {c2})^2}{\text {b2}}+\text {b2} c_1}}{\sqrt {\frac {1}{\text {b2}}}}+\text {b1} x+\text {c2}}{\text {b2}} \\ y(x)\to -\frac {\text {b1} x+\text {c2}}{\text {b2}}+\sqrt {\frac {1}{\text {b2}}} \sqrt {-x (\text {a1} x+2 \text {c1})+\frac {(\text {b1} x+\text {c2})^2}{\text {b2}}+\text {b2} c_1} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a__1 = symbols("a__1") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
c__1 = symbols("c__1") 
c__2 = symbols("c__2") 
y = Function("y") 
ode = Eq(a__1*x + b__1*y(x) + c__1 + (b__1*x + b__2*y(x) + c__2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-a__1*x - b__1*y(x) - c__1)/(b__1*x + b__2*y(x) + c__2) + Derivative(y(x), x) cannot be solved by the factorable group method