29.15.17 problem 425
Internal
problem
ID
[5023]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
15
Problem
number
:
425
Date
solved
:
Sunday, March 30, 2025 at 06:30:23 AM
CAS
classification
:
[_Bernoulli]
\begin{align*} y y^{\prime }&=b \cos \left (x +c \right )+a y^{2} \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 106
ode:=y(x)*diff(y(x),x) = b*cos(x+c)+a*y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {16 \left (a^{2}+\frac {1}{4}\right )^{2} c_1 \,{\mathrm e}^{2 a x}-16 \left (a^{2}+\frac {1}{4}\right ) \left (a \cos \left (x +c \right )-\frac {\sin \left (x +c \right )}{2}\right ) b}}{4 a^{2}+1} \\
y &= -\frac {\sqrt {16 \left (a^{2}+\frac {1}{4}\right )^{2} c_1 \,{\mathrm e}^{2 a x}-16 \left (a^{2}+\frac {1}{4}\right ) \left (a \cos \left (x +c \right )-\frac {\sin \left (x +c \right )}{2}\right ) b}}{4 a^{2}+1} \\
\end{align*}
✓ Mathematica. Time used: 5.731 (sec). Leaf size: 106
ode=y[x] D[y[x],x]== b Cos[x+c]+a y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {\left (4 a^2+1\right ) c_1 e^{2 a x}-4 a b \cos (c+x)+2 b \sin (c+x)}}{\sqrt {4 a^2+1}} \\
y(x)\to \frac {\sqrt {\left (4 a^2+1\right ) c_1 e^{2 a x}-4 a b \cos (c+x)+2 b \sin (c+x)}}{\sqrt {4 a^2+1}} \\
\end{align*}
✓ Sympy. Time used: 112.309 (sec). Leaf size: 430
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a*y(x)**2 - b*cos(c + x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {C_{1} e^{2 a x} - i b x e^{2 a x + i x} \sin {\left (c + x \right )} + b x e^{2 a x + i x} \cos {\left (c + x \right )} + b e^{2 a x + i x} \sin {\left (c + x \right )}} & \text {for}\: a = - \frac {i}{2} \\- \sqrt {C_{1} e^{2 a x} + i b x e^{2 a x - i x} \sin {\left (c + x \right )} + b x e^{2 a x - i x} \cos {\left (c + x \right )} + b e^{2 a x - i x} \sin {\left (c + x \right )}} & \text {for}\: a = \frac {i}{2} \\- \sqrt {\frac {4 C_{1} a^{2} e^{2 a x}}{4 a^{2} + 1} + \frac {C_{1} e^{2 a x}}{4 a^{2} + 1} - \frac {4 a b \cos {\left (c + x \right )}}{4 a^{2} + 1} + \frac {2 b \sin {\left (c + x \right )}}{4 a^{2} + 1}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {C_{1} e^{2 a x} - i b x e^{2 a x + i x} \sin {\left (c + x \right )} + b x e^{2 a x + i x} \cos {\left (c + x \right )} + b e^{2 a x + i x} \sin {\left (c + x \right )}} & \text {for}\: a = - \frac {i}{2} \\\sqrt {C_{1} e^{2 a x} + i b x e^{2 a x - i x} \sin {\left (c + x \right )} + b x e^{2 a x - i x} \cos {\left (c + x \right )} + b e^{2 a x - i x} \sin {\left (c + x \right )}} & \text {for}\: a = \frac {i}{2} \\\sqrt {\frac {4 C_{1} a^{2} e^{2 a x}}{4 a^{2} + 1} + \frac {C_{1} e^{2 a x}}{4 a^{2} + 1} - \frac {4 a b \cos {\left (c + x \right )}}{4 a^{2} + 1} + \frac {2 b \sin {\left (c + x \right )}}{4 a^{2} + 1}} & \text {otherwise} \end {cases}\right ]
\]