23.1.140 problem 143 (b)
Internal
problem
ID
[4747]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
143
(b)
Date
solved
:
Friday, October 03, 2025 at 01:21:51 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} 2 y^{\prime }+a x&=-\sqrt {a^{2} x^{2}-4 b \,x^{2}-4 c y} \end{align*}
✓ Maple. Time used: 0.139 (sec). Leaf size: 420
ode:=2*diff(y(x),x)+a*x = -(a^2*x^2-4*b*x^2-4*c*y(x))^(1/2);
dsolve(ode,y(x), singsol=all);
\[
-\int _{\textit {\_b}}^{x}\frac {a \textit {\_a} +\sqrt {-4 c y+\left (a^{2}-4 b \right ) \textit {\_a}^{2}}}{a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {-4 c y+\left (a^{2}-4 b \right ) \textit {\_a}^{2}}+4 y}d \textit {\_a} -2 \int _{}^{y}\frac {2 \int _{\textit {\_b}}^{x}\frac {\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} a \,x^{2}+2 \int _{\textit {\_b}}^{x}\frac {\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} x \sqrt {\left (a^{2}-4 b \right ) x^{2}-4 \textit {\_f} c}+8 \int _{\textit {\_b}}^{x}\frac {\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}+\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} \textit {\_f} +1}{a \,x^{2}+x \sqrt {\left (a^{2}-4 b \right ) x^{2}-4 \textit {\_f} c}+4 \textit {\_f}}d \textit {\_f} +c_1 = 0
\]
✓ Mathematica. Time used: 0.372 (sec). Leaf size: 542
ode=2 D[y[x],x]+a*x==-Sqrt[a^2*x^2-4*b*x^2 -4*c*y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4-2 \text {$\#$1}^3 c-2 \text {$\#$1}^2 a^2-4 \text {$\#$1}^2 a c+8 \text {$\#$1}^2 b-2 \text {$\#$1} a^2 c+8 \text {$\#$1} b c+a^4-8 a^2 b+16 b^2\&,\frac {-\text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+\text {$\#$1}^3 \log (x)+\text {$\#$1}^2 c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-\text {$\#$1}^2 c \log (x)+\text {$\#$1} a^2 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-4 \text {$\#$1} b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+2 \text {$\#$1} a c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+a^2 c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-4 b c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-\text {$\#$1} a^2 \log (x)-2 \text {$\#$1} a c \log (x)+4 \text {$\#$1} b \log (x)-a^2 c \log (x)+4 b c \log (x)}{-2 \text {$\#$1}^3+3 \text {$\#$1}^2 c+2 \text {$\#$1} a^2+4 \text {$\#$1} a c-8 \text {$\#$1} b+a^2 c-4 b c}\&\right ]-\log \left (\sqrt {-c y(x)} \sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 c y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*x + sqrt(a**2*x**2 - 4*b*x**2 - 4*c*y(x)) + 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out