54.1.465 problem 478

Internal problem ID [11779]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 478
Date solved : Tuesday, September 30, 2025 at 10:21:22 PM
CAS classification : [_quadrature]

\begin{align*} \left (a y+b \right ) \left ({y^{\prime }}^{2}+1\right )-c&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 162

ode:=(a*y(x)+b)*(1+diff(y(x),x)^2)-c = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-b +c}{a} \\ \frac {-\arctan \left (\frac {2 a y+2 b -c}{2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}}\right ) c +2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}+\left (2 x -2 c_1 \right ) a}{2 a} &= 0 \\ \frac {\arctan \left (\frac {2 a y+2 b -c}{2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}}\right ) c -2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}+\left (2 x -2 c_1 \right ) a}{2 a} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.56 (sec). Leaf size: 240

ode=-c + (b + a*y[x])*(1 + D[y[x],x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 3.951 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c + (a*y(x) + b)*(Derivative(y(x), x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {y a + b - c}{y a + b}}}\, dy = C_{1} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {y a + b - c}{y a + b}}}\, dy = C_{1} - x\right ] \]