23.1.438 problem 428
Internal
problem
ID
[5045]
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
:
428
Date
solved
:
Tuesday, September 30, 2025 at 11:29:53 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: 0.229 (sec). Leaf size: 82
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 -e^{a x} \sqrt {2 \int _1^xb e^{-2 a K[1]} \cos (c+K[1])dK[1]+c_1}\\ y(x)&\to e^{a x} \sqrt {2 \int _1^xb e^{-2 a K[1]} \cos (c+K[1])dK[1]+c_1} \end{align*}
✓ Sympy. Time used: 59.140 (sec). Leaf size: 437
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)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {C_{1} e^{2 a x} - i b x e^{2 a x + i x} \sin {\left (c + x \right )} + b x e^{2 a x + i x} \cos {\left (c + x \right )} - i b e^{2 a x + i x} \cos {\left (c + x \right )}} & \text {for}\: a = - \frac {i}{2} \\- \sqrt {C_{1} e^{2 a x} + i b x e^{2 a x - i x} \sin {\left (c + x \right )} + b x e^{2 a x - i x} \cos {\left (c + x \right )} + i b e^{2 a x - i x} \cos {\left (c + x \right )}} & \text {for}\: a = \frac {i}{2} \\- \sqrt {\frac {4 C_{1} a^{2} e^{2 a x}}{4 a^{2} + 1} + \frac {C_{1} e^{2 a x}}{4 a^{2} + 1} - \frac {4 a b \cos {\left (c + x \right )}}{4 a^{2} + 1} + \frac {2 b \sin {\left (c + x \right )}}{4 a^{2} + 1}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {C_{1} e^{2 a x} - i b x e^{2 a x + i x} \sin {\left (c + x \right )} + b x e^{2 a x + i x} \cos {\left (c + x \right )} - i b e^{2 a x + i x} \cos {\left (c + x \right )}} & \text {for}\: a = - \frac {i}{2} \\\sqrt {C_{1} e^{2 a x} + i b x e^{2 a x - i x} \sin {\left (c + x \right )} + b x e^{2 a x - i x} \cos {\left (c + x \right )} + i b e^{2 a x - i x} \cos {\left (c + x \right )}} & \text {for}\: a = \frac {i}{2} \\\sqrt {\frac {4 C_{1} a^{2} e^{2 a x}}{4 a^{2} + 1} + \frac {C_{1} e^{2 a x}}{4 a^{2} + 1} - \frac {4 a b \cos {\left (c + x \right )}}{4 a^{2} + 1} + \frac {2 b \sin {\left (c + x \right )}}{4 a^{2} + 1}} & \text {otherwise} \end {cases}\right ]
\]