60.1.456 problem 469
Internal
problem
ID
[10470]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
469
Date
solved
:
Sunday, March 30, 2025 at 04:50:59 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}+a x y^{\prime }+b y&=0 \end{align*}
✓ Maple. Time used: 0.134 (sec). Leaf size: 108
ode:=y(x)*diff(y(x),x)^2+a*x*diff(y(x),x)+b*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {-4 \textit {\_a}^{2} b +a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {2 \textit {\_a}^{2}+a -\sqrt {-4 \textit {\_a}^{2} b +a^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 0.646 (sec). Leaf size: 162
ode=b*y[x] + a*x*D[y[x],x] + y[x]*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}-a-2 b\right )}{4 (a+b)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}-a\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )}{4 (a+b)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 76.613 (sec). Leaf size: 214
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*x*Derivative(y(x), x) + b*y(x) + y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} + \begin {cases} - \log {\left (\left (\left (a + \sqrt {a^{2} - \frac {4 b y^{2}{\left (x \right )}}{x^{2}}}\right )^{\frac {a b}{2 \left (a + b\right )}} \left (a + 2 b - \sqrt {a^{2} - \frac {4 b y^{2}{\left (x \right )}}{x^{2}}}\right )^{\frac {b \left (a + 2 b\right )}{2 \left (a + b\right )}}\right )^{\frac {1}{b}} \right )} & \text {for}\: b \neq 0 \\- \log {\left (\sqrt {2 a + 2 \sqrt {a^{2}} + \frac {4 y^{2}{\left (x \right )}}{x^{2}}} \right )} & \text {otherwise} \end {cases}, \ \log {\left (x \right )} = C_{1} + \begin {cases} - \log {\left (\left (\left (a - \sqrt {a^{2} - \frac {4 b y^{2}{\left (x \right )}}{x^{2}}}\right )^{\frac {a b}{2 \left (a + b\right )}} \left (a + 2 b + \sqrt {a^{2} - \frac {4 b y^{2}{\left (x \right )}}{x^{2}}}\right )^{\frac {b \left (a + 2 b\right )}{2 \left (a + b\right )}}\right )^{\frac {1}{b}} \right )} & \text {for}\: b \neq 0 \\- \log {\left (\sqrt {2 a - 2 \sqrt {a^{2}} + \frac {4 y^{2}{\left (x \right )}}{x^{2}}} \right )} & \text {otherwise} \end {cases}\right ]
\]