60.1.77 problem 78
Internal
problem
ID
[10091]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
78
Date
solved
:
Sunday, March 30, 2025 at 03:15:20 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b&=0 \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 88
ode:=diff(y(x),x)+a*sin(alpha*y(x)+beta*x)+b = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\beta x +2 \arctan \left (\frac {\tan \left (\frac {\sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}\, \left (c_1 -x \right )}{2}\right ) \sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}-a \alpha }{b \alpha -\beta }\right )}{\alpha }
\]
✓ Mathematica. Time used: 0.957 (sec). Leaf size: 396
ode=D[y[x],x]+ a*Sin[\[Alpha]*y[x]+\[Beta]*x] + b==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}-\frac {b \int _1^x\left (\frac {a \alpha ^2 \cos (\beta K[1]+\alpha K[2])}{b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta }-\frac {a \alpha ^3 \cos (\beta K[1]+\alpha K[2]) (b+a \sin (\beta K[1]+\alpha K[2]))}{(b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta )^2}\right )dK[1] \alpha +a \sin (x \beta +\alpha K[2]) \int _1^x\left (\frac {a \alpha ^2 \cos (\beta K[1]+\alpha K[2])}{b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta }-\frac {a \alpha ^3 \cos (\beta K[1]+\alpha K[2]) (b+a \sin (\beta K[1]+\alpha K[2]))}{(b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta )^2}\right )dK[1] \alpha -\alpha -\beta \int _1^x\left (\frac {a \alpha ^2 \cos (\beta K[1]+\alpha K[2])}{b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta }-\frac {a \alpha ^3 \cos (\beta K[1]+\alpha K[2]) (b+a \sin (\beta K[1]+\alpha K[2]))}{(b \alpha +a \sin (\beta K[1]+\alpha K[2]) \alpha -\beta )^2}\right )dK[1]}{b \alpha +a \sin (x \beta +\alpha K[2]) \alpha -\beta }dK[2]+\int _1^x\frac {\alpha (b+a \sin (\beta K[1]+\alpha y(x)))}{b \alpha +a \sin (\beta K[1]+\alpha y(x)) \alpha -\beta }dK[1]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*sin(Alpha*y(x) + BETA*x) + b + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out