54.1.35 problem 35

Internal problem ID [11349]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 08:03:43 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right )&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 35

ode:=diff(y(x),x)+f(x)*(y(x)^2+2*a*y(x)+b) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -a +\tanh \left (\sqrt {a^{2}-b}\, \left (\int f \left (x \right )d x +c_1 \right )\right ) \sqrt {a^{2}-b} \]

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 83

ode=D[y[x],x] + f[x]*(y[x]^2 + 2*a*y[x] +b)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+2 a K[1]+b}dK[1]\&\right ]\left [\int _1^x-f(K[2])dK[2]+c_1\right ]\\ y(x)&\to -\sqrt {a^2-b}-a\\ y(x)&\to \sqrt {a^2-b}-a \end{align*}

✓ Sympy. Time used: 1.174 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((2*a*y(x) + b + y(x)**2)*f(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \frac {\sqrt {\frac {1}{a^{2} - b}} \log {\left (- a^{2} \sqrt {\frac {1}{a^{2} - b}} + a + b \sqrt {\frac {1}{a^{2} - b}} + y{\left (x \right )} \right )}}{2} - \frac {\sqrt {\frac {1}{a^{2} - b}} \log {\left (a^{2} \sqrt {\frac {1}{a^{2} - b}} + a - b \sqrt {\frac {1}{a^{2} - b}} + y{\left (x \right )} \right )}}{2} + \int f{\left (x \right )}\, dx = C_{1} \]