29.4.10 problem 99
Internal
problem
ID
[4701]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
4
Problem
number
:
99
Date
solved
:
Sunday, March 30, 2025 at 03:40:02 AM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }&=a +b y+\sqrt {\operatorname {A0} +\operatorname {B0} y} \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 27
ode:=diff(y(x),x) = a+b*y(x)+(A0+B0*y(x))^(1/2);
dsolve(ode,y(x), singsol=all);
\[
x -\int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} +c_1 = 0
\]
✓ Mathematica. Time used: 0.62 (sec). Leaf size: 199
ode=D[y[x],x]==a+b y[x]+Sqrt[A0+B0 y[x]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {2 \text {B0} \sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2} \text {arctanh}\left (\frac {2 b \sqrt {\text {$\#$1} \text {B0}+\text {A0}}+\text {B0}}{\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}}\right )}{b \left (\text {B0}^2-4 b (a \text {B0}-\text {A0} b)\right )}+\frac {\log \left (-b (\text {$\#$1} \text {B0}+\text {A0})-\text {B0} \sqrt {\text {$\#$1} \text {B0}+\text {A0}}-a \text {B0}+\text {A0} b\right )}{b}\&\right ][x+c_1] \\
y(x)\to -\frac {\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}+2 a b-\text {B0}}{2 b^2} \\
y(x)\to \frac {\sqrt {-4 a b \text {B0}+4 \text {A0} b^2+\text {B0}^2}-2 a b+\text {B0}}{2 b^2} \\
\end{align*}
✓ Sympy. Time used: 4.043 (sec). Leaf size: 204
from sympy import *
x = symbols("x")
A0 = symbols("A0")
B0 = symbols("B0")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a - b*y(x) - sqrt(A0 + B0*y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\begin {cases} \frac {2 B_{0} \operatorname {atan}{\left (\frac {2 \left (\frac {B_{0}}{2 b} + \sqrt {A_{0} + B_{0} y{\left (x \right )}}\right )}{\sqrt {- \frac {4 A_{0} b^{2} + B_{0}^{2} - 4 B_{0} a b}{b^{2}}}} \right )}}{b^{2} \sqrt {- \frac {4 A_{0} b^{2} + B_{0}^{2} - 4 B_{0} a b}{b^{2}}}} - \frac {\log {\left (- A_{0} b + B_{0} a + B_{0} \sqrt {A_{0} + B_{0} y{\left (x \right )}} + b \left (A_{0} + B_{0} y{\left (x \right )}\right ) \right )}}{b} & \text {for}\: B_{0} \neq 0 \wedge 4 A_{0} b^{2} + B_{0}^{2} - 4 B_{0} a b \neq 0 \\- \frac {B_{0}}{b^{2} \left (\frac {B_{0}}{2 b} + \sqrt {A_{0} + B_{0} y{\left (x \right )}}\right )} - \frac {\log {\left (- A_{0} b + B_{0} a + B_{0} \sqrt {A_{0} + B_{0} y{\left (x \right )}} + b \left (A_{0} + B_{0} y{\left (x \right )}\right ) \right )}}{b} & \text {for}\: B_{0} \neq 0 \\- \frac {y{\left (x \right )}}{\sqrt {A_{0}} + a} & \text {for}\: b = 0 \\- \frac {\log {\left (- \sqrt {A_{0}} - a - b y{\left (x \right )} \right )}}{b} & \text {otherwise} \end {cases} = C_{1} - x
\]