71.1.29 problem 46 (page 56)
Internal
problem
ID
[19205]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
46
(page
56)
Date
solved
:
Thursday, October 02, 2025 at 03:43:22 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} \left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }&=\left (1+y^{2}\right )^{{3}/{2}} \end{align*}
✓ Maple. Time used: 0.201 (sec). Leaf size: 88
ode:=(y(x)-x)*(x^2+1)^(1/2)*diff(y(x),x) = (1+y(x)^2)^(3/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {\arctan \left (y\right ) y x -\sqrt {\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y+1\right )^{2}}}\, y x -y c_1 x +\arctan \left (y\right )-\sqrt {\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y+1\right )^{2}}}-y-c_1 +x}{x y+1} = 0
\]
✓ Mathematica. Time used: 1.888 (sec). Leaf size: 592
ode=(y[x]-x)*Sqrt[1+x^2]*D[y[x],x]==(1+y[x]^2)^(3/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2+1} x}{\sqrt {x^2+1} (x K[2]+1)}+\frac {\sqrt {x^2+1} \sqrt {K[2]^2+1} x}{(x K[2]+1)^2}-\frac {x}{\sqrt {x^2+1} \sqrt {K[2]^2+1}}-\int _1^x\left (\frac {\sqrt {K[1]^2+1} K[2]^2}{(K[1] K[2]+1)^2 \sqrt {K[2]^2+1}}+\frac {K[2]^2}{\sqrt {K[1]^2+1} \left (K[2]^2+1\right )^{3/2}}-\frac {\sqrt {K[1]^2+1} K[2]^2}{(K[1] K[2]+1) \left (K[2]^2+1\right )^{3/2}}-\frac {2 K[1] \sqrt {K[1]^2+1} \sqrt {K[2]^2+1} K[2]}{(K[1] K[2]+1)^3}-\frac {K[1] \sqrt {K[1]^2+1} K[2]}{(K[1] K[2]+1)^2 \sqrt {K[2]^2+1}}+\frac {K[1] K[2]}{\sqrt {K[1]^2+1} \left (K[2]^2+1\right )^{3/2}}+\frac {2 K[2]}{(K[1] K[2]+1)^2}-\frac {2 K[1] \left (K[2]^2+1\right )}{(K[1] K[2]+1)^3}+\frac {\sqrt {K[1]^2+1} \sqrt {K[2]^2+1}}{(K[1] K[2]+1)^2}-\frac {1}{\sqrt {K[1]^2+1} \sqrt {K[2]^2+1}}+\frac {\sqrt {K[1]^2+1}}{(K[1] K[2]+1) \sqrt {K[2]^2+1}}\right )dK[1]-\frac {K[2]}{\sqrt {x^2+1} \sqrt {K[2]^2+1}}+\frac {1}{K[2]^2+1}+\frac {-x^2-1}{(x K[2]+1)^2}\right )dK[2]+\int _1^x\left (-\frac {K[1]}{\sqrt {K[1]^2+1} \sqrt {y(x)^2+1}}+\frac {y(x)^2+1}{(K[1] y(x)+1)^2}+\frac {\sqrt {K[1]^2+1} y(x) \sqrt {y(x)^2+1}}{(K[1] y(x)+1)^2}-\frac {y(x)}{\sqrt {K[1]^2+1} \sqrt {y(x)^2+1}}+\frac {\sqrt {K[1]^2+1} y(x)}{(K[1] y(x)+1) \sqrt {y(x)^2+1}}\right )dK[1]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x + y(x))*sqrt(x**2 + 1)*Derivative(y(x), x) - (y(x)**2 + 1)**(3/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out