75.12.2 problem 276

Internal problem ID [16797]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 276
Date solved : Monday, March 31, 2025 at 03:19:47 PM
CAS classification : [_linear]

\begin{align*} x \sin \left (x \right ) y^{\prime }+\left (\sin \left (x \right )-x \cos \left (x \right )\right ) y&=\sin \left (x \right ) \cos \left (x \right )-x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

ode:=x*sin(x)*diff(y(x),x)+(sin(x)-x*cos(x))*y(x) = cos(x)*sin(x)-x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\frac {x}{\tan \left (x \right )}-\ln \left (\tan \left (x \right )\right )+\frac {\ln \left (\sec \left (x \right )^{2}\right )}{2}+\frac {\ln \left (-1+\cos \left (2 x \right )\right )}{2}+c_1 \right ) \sin \left (x \right )}{x} \]

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 34

ode=x*Sin[x]*D[y[x],x]+(Sin[x]-x*Cos[x])*y[x]==Sin[x]*Cos[x]-x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\sin (x) \left (\int _1^x\left (\cot (K[1])-\csc ^2(K[1]) K[1]\right )dK[1]+c_1\right )}{x} \]

✓ Sympy. Time used: 21.760 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*sin(x)*Derivative(y(x), x) + x + (-x*cos(x) + sin(x))*y(x) - sin(x)*cos(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \frac {\sqrt {\tan ^{2}{\left (x \right )} + 1} y{\left (x \right )}}{\tan {\left (x \right )}}\, dx - \int \frac {\sqrt {\tan ^{2}{\left (x \right )} + 1} \cos {\left (x \right )}}{\tan {\left (x \right )}}\, dx - \int \frac {x \sqrt {\tan ^{2}{\left (x \right )} + 1} y{\left (x \right )}}{\tan ^{2}{\left (x \right )}}\, dx + \int \frac {x \sqrt {\tan ^{2}{\left (x \right )} + 1}}{\sin {\left (x \right )} \tan {\left (x \right )}}\, dx = C_{1} \]