77.8.15 problem 16
Internal
problem
ID
[20426]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Misc
examples
on
chapter
II
at
page
25
Problem
number
:
16
Date
solved
:
Thursday, October 02, 2025 at 05:58:37 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y y^{\prime }+b y^{2}&=a \cos \left (x \right ) \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 98
ode:=y(x)*diff(y(x),x)+b*y(x)^2 = a*cos(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {16 c_1 \left (b^{2}+\frac {1}{4}\right )^{2} {\mathrm e}^{-2 b x}+16 \left (\cos \left (x \right ) b +\frac {\sin \left (x \right )}{2}\right ) a \left (b^{2}+\frac {1}{4}\right )}}{4 b^{2}+1} \\
y &= -\frac {\sqrt {16 c_1 \left (b^{2}+\frac {1}{4}\right )^{2} {\mathrm e}^{-2 b x}+16 \left (\cos \left (x \right ) b +\frac {\sin \left (x \right )}{2}\right ) a \left (b^{2}+\frac {1}{4}\right )}}{4 b^{2}+1} \\
\end{align*}
✓ Mathematica. Time used: 5.306 (sec). Leaf size: 112
ode=y[x]*D[y[x],x]+b*y[x]^2==a*Cos[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt {4 a b \cos (x)+e^{-2 b x} \left (2 a e^{2 b x} \sin (x)+4 b^2 c_1+c_1\right )}}{\sqrt {4 b^2+1}}\\ y(x)&\to \frac {\sqrt {4 a b \cos (x)+e^{-2 b x} \left (2 a e^{2 b x} \sin (x)+4 b^2 c_1+c_1\right )}}{\sqrt {4 b^2+1}} \end{align*}
✓ Sympy. Time used: 29.270 (sec). Leaf size: 508
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*cos(x) + b*y(x)**2 + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {C_{1} e^{- 2 b x} + i a x e^{- 2 b x - i x} \sin {\left (x \right )} + a x e^{- 2 b x - i x} \cos {\left (x \right )} + a e^{- 2 b x - i x} \sin {\left (x \right )}} & \text {for}\: b = - \frac {i}{2} \\- \sqrt {C_{1} e^{- 2 b x} - i a x e^{- 2 b x + i x} \sin {\left (x \right )} + a x e^{- 2 b x + i x} \cos {\left (x \right )} + a e^{- 2 b x + i x} \sin {\left (x \right )}} & \text {for}\: b = \frac {i}{2} \\- \sqrt {\frac {4 C_{1} b^{2}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {C_{1}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {4 a b e^{2 b x} \cos {\left (x \right )}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {2 a e^{2 b x} \sin {\left (x \right )}}{4 b^{2} e^{2 b x} + e^{2 b x}}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {C_{1} e^{- 2 b x} + i a x e^{- 2 b x - i x} \sin {\left (x \right )} + a x e^{- 2 b x - i x} \cos {\left (x \right )} + a e^{- 2 b x - i x} \sin {\left (x \right )}} & \text {for}\: b = - \frac {i}{2} \\\sqrt {C_{1} e^{- 2 b x} - i a x e^{- 2 b x + i x} \sin {\left (x \right )} + a x e^{- 2 b x + i x} \cos {\left (x \right )} + a e^{- 2 b x + i x} \sin {\left (x \right )}} & \text {for}\: b = \frac {i}{2} \\\sqrt {\frac {4 C_{1} b^{2}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {C_{1}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {4 a b e^{2 b x} \cos {\left (x \right )}}{4 b^{2} e^{2 b x} + e^{2 b x}} + \frac {2 a e^{2 b x} \sin {\left (x \right )}}{4 b^{2} e^{2 b x} + e^{2 b x}}} & \text {otherwise} \end {cases}\right ]
\]