77.1.55 problem 74 (page 112)
Internal
problem
ID
[17874]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
74
(page
112)
Date
solved
:
Monday, March 31, 2025 at 04:43:47 PM
CAS
classification
:
[_quadrature]
\begin{align*} y \left (1+{y^{\prime }}^{2}\right )&=2 \alpha \end{align*}
✓ Maple. Time used: 0.050 (sec). Leaf size: 325
ode:=y(x)*(1+diff(y(x),x)^2) = 2*alpha;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 1.302 (sec). Leaf size: 356
ode=y[x]*(1+D[y[x],x]^2)==a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {a-\text {$\#$1}}-\sqrt {a-1}\right ) \left (\text {$\#$1} a+\sqrt {a-1} a \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}} a-2 \sqrt {\text {$\#$1}} \sqrt {a-1} \sqrt {a-\text {$\#$1}}-2 \text {$\#$1}-a^2+a\right )}{\left (\sqrt {a-1} \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}}-a\right )^2}+2 a \arctan \left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {a-1}-\sqrt {a-\text {$\#$1}}}\right )\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {a-\text {$\#$1}}-\sqrt {a-1}\right ) \left (\text {$\#$1} a+\sqrt {a-1} a \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}} a-2 \sqrt {\text {$\#$1}} \sqrt {a-1} \sqrt {a-\text {$\#$1}}-2 \text {$\#$1}-a^2+a\right )}{\left (\sqrt {a-1} \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}}-a\right )^2}+2 a \arctan \left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {a-1}-\sqrt {a-\text {$\#$1}}}\right )\&\right ][x+c_1] \\
y(x)\to a \\
\end{align*}
✓ Sympy. Time used: 1.832 (sec). Leaf size: 296
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
y = Function("y")
ode = Eq(-2*Alpha + (Derivative(y(x), x)**2 + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - \sqrt {2} i \sqrt {\mathrm {A}} \sqrt {-1 + \frac {y{\left (x \right )}}{2 \mathrm {A}}} \sqrt {y{\left (x \right )}} - 2 i \mathrm {A} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2 \sqrt {\mathrm {A}}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{\mathrm {A}}}\right | > 2 \\- \frac {\sqrt {2} \sqrt {\mathrm {A}} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{2 \mathrm {A}}}} + 2 \mathrm {A} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2 \sqrt {\mathrm {A}}} \right )} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {\mathrm {A}} \sqrt {1 - \frac {y{\left (x \right )}}{2 \mathrm {A}}}} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} - \sqrt {2} i \sqrt {\mathrm {A}} \sqrt {-1 + \frac {y{\left (x \right )}}{2 \mathrm {A}}} \sqrt {y{\left (x \right )}} - 2 i \mathrm {A} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2 \sqrt {\mathrm {A}}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{\mathrm {A}}}\right | > 2 \\- \frac {\sqrt {2} \sqrt {\mathrm {A}} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{2 \mathrm {A}}}} + 2 \mathrm {A} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2 \sqrt {\mathrm {A}}} \right )} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {\mathrm {A}} \sqrt {1 - \frac {y{\left (x \right )}}{2 \mathrm {A}}}} & \text {otherwise} \end {cases} = C_{1} + x\right ]
\]