29.9.1 problem 241
Internal
problem
ID
[4841]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
9
Problem
number
:
241
Date
solved
:
Sunday, March 30, 2025 at 04:03:22 AM
CAS
classification
:
[_separable]
\begin{align*} 2 x y^{\prime }+4 y+a +\sqrt {a^{2}-4 b -4 c y}&=0 \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 34
ode:=2*x*diff(y(x),x)+4*y(x)+a+(a^2-4*b-4*c*y(x))^(1/2) = 0;
dsolve(ode,y(x), singsol=all);
\[
\ln \left (x \right )+2 \int _{}^{y}\frac {1}{4 \textit {\_a} +a +\sqrt {-4 \textit {\_a} c +a^{2}-4 b}}d \textit {\_a} +c_1 = 0
\]
✓ Mathematica. Time used: 0.883 (sec). Leaf size: 176
ode=2 x D[y[x],x]+4 y[x]+a +Sqrt[a^2-4 b- 4 c y[x]]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {1}{4} \left (\log \left (-c \left (\sqrt {a^2-4 (\text {$\#$1} c+b)}+4 \text {$\#$1}+a\right )\right )-\frac {2 c \text {arctanh}\left (\frac {2 \sqrt {a^2-4 (\text {$\#$1} c+b)}-c}{\sqrt {4 a^2+4 a c-16 b+c^2}}\right )}{\sqrt {4 a^2+4 a c-16 b+c^2}}\right )\&\right ]\left [-\frac {\log (x)}{2}+c_1\right ] \\
y(x)\to \frac {1}{8} \left (-\sqrt {(2 a+c)^2-16 b}-2 a-c\right ) \\
y(x)\to \frac {1}{8} \left (\sqrt {(2 a+c)^2-16 b}-2 a-c\right ) \\
\end{align*}
✓ Sympy. Time used: 4.064 (sec). Leaf size: 192
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a + 2*x*Derivative(y(x), x) + sqrt(a**2 - 4*b - 4*c*y(x)) + 4*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\begin {cases} \frac {c \operatorname {atan}{\left (\frac {- \frac {c}{2} + \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}}}{\sqrt {- a^{2} - a c + 4 b - \frac {c^{2}}{4}}} \right )}}{4 \sqrt {- a^{2} - a c + 4 b - \frac {c^{2}}{4}}} + \frac {\log {\left (a c + c \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}} + 4 c y{\left (x \right )} \right )}}{4} & \text {for}\: c \neq 0 \wedge 4 a^{2} + 4 a c - 16 b + c^{2} \neq 0 \\- \frac {c}{4 \left (- \frac {c}{2} + \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}}\right )} + \frac {\log {\left (a c + c \sqrt {a^{2} - 4 b - 4 c y{\left (x \right )}} + 4 c y{\left (x \right )} \right )}}{4} & \text {for}\: c \neq 0 \\\frac {\log {\left (a + \sqrt {a^{2} - 4 b} + 4 y{\left (x \right )} \right )}}{4} & \text {otherwise} \end {cases} = C_{1} - \frac {\log {\left (x \right )}}{2}
\]