89.10.9 problem 9

Internal problem ID [24502]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 77
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:43:21 PM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.149 (sec). Leaf size: 83

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

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

✓ Mathematica. Time used: 19.694 (sec). Leaf size: 106

ode=( a1*x+k*y[x]+c1)+( k*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 {c2}+k x)^2}{\text {b2}}+\text {b2} c_1}}{\sqrt {\frac {1}{\text {b2}}}}+\text {c2}+k x}{\text {b2}}\\ y(x)&\to -\frac {\text {c2}+k x}{\text {b2}}+\sqrt {\frac {1}{\text {b2}}} \sqrt {-x (\text {a1} x+2 \text {c1})+\frac {(\text {c2}+k x)^2}{\text {b2}}+\text {b2} c_1} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
k = symbols("k") 
c1 = symbols("c1") 
b2 = symbols("b2") 
c2 = symbols("c2") 
y = Function("y") 
ode = Eq(a1*x + c1 + k*y(x) + (b2*y(x) + c2 + k*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-a1*x - c1 - k*y(x))/(b2*y(x) + c2 + k*x) + Derivative(y(x), x