54.1.419 problem 430

Internal problem ID [11733]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 430
Date solved : Tuesday, September 30, 2025 at 10:14:48 PM
CAS classification : [_dAlembert]

\begin{align*} \left (\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0}&=0 \end{align*}

✗ Maple

ode:=(a2*x+c2)*diff(y(x),x)^2+(x*a1+b1*y(x)+c1)*diff(y(x),x)+a0*x+b0*y(x)+c0 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 171.217 (sec). Leaf size: 416

ode=c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*D[y[x],x] + (c2 + a2*x)*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=\exp \left (\int _1^{K[1]}\frac {\frac {-\text {a1}-2 \text {a2} K[2]}{\text {b0}+\text {b1} K[2]}-\frac {\text {b1} \left (-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}\right )}{(\text {b0}+\text {b1} K[2])^2}}{K[2]-\frac {-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}}{\text {b0}+\text {b1} K[2]}}dK[2]\right ) \int \frac {\left (\frac {-2 \text {c2} K[1]-\text {c1}}{\text {b1} K[1]+\text {b0}}-\frac {\text {b1} \left (-\text {c1} K[1]-\text {c2} K[1]^2-\text {c0}\right )}{(\text {b1} K[1]+\text {b0})^2}\right ) \exp \left (-\int _1^{K[1]}\frac {\frac {-\text {a1}-2 \text {a2} K[2]}{\text {b0}+\text {b1} K[2]}-\frac {\text {b1} \left (-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}\right )}{(\text {b0}+\text {b1} K[2])^2}}{K[2]-\frac {-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}}{\text {b0}+\text {b1} K[2]}}dK[2]\right )}{K[1]-\frac {-\text {a1} K[1]-\text {a2} K[1]^2-\text {a0}}{\text {b1} K[1]+\text {b0}}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}\frac {\frac {-\text {a1}-2 \text {a2} K[2]}{\text {b0}+\text {b1} K[2]}-\frac {\text {b1} \left (-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}\right )}{(\text {b0}+\text {b1} K[2])^2}}{K[2]-\frac {-\text {a2} K[2]^2-\text {a1} K[2]-\text {a0}}{\text {b0}+\text {b1} K[2]}}dK[2]\right ),y(x)=\frac {x \left (-\text {a1} K[1]-\text {a2} K[1]^2-\text {a0}\right )}{\text {b1} K[1]+\text {b0}}+\frac {-\text {c1} K[1]-\text {c2} K[1]^2-\text {c0}}{\text {b1} K[1]+\text {b0}}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b0 = symbols("b0") 
b1 = symbols("b1") 
c0 = symbols("c0") 
c1 = symbols("c1") 
c2 = symbols("c2") 
y = Function("y") 
ode = Eq(a0*x + b0*y(x) + c0 + (a2*x + c2)*Derivative(y(x), x)**2 + (a1*x + b1*y(x) + c1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out