60.1.465 problem 478
Internal
problem
ID
[10479]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
478
Date
solved
:
Sunday, March 30, 2025 at 04:51:23 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: 0.679 (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: 5.020 (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)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {y a + b - c}{y a + b}}}\, dy = C_{1} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {y a + b - c}{y a + b}}}\, dy = C_{1} - x\right ]
\]