63.1.79 problem 124

Internal problem ID [15519]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 124
Date solved : Thursday, October 02, 2025 at 10:19:23 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }&=\sin \left (2 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+tan(x)*diff(y(x),x) = sin(2*x); 
ic:=[y(0) = -1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -x -1+2 \sin \left (x \right )-\frac {\sin \left (2 x \right )}{2} \]

✓ Mathematica. Time used: 28.245 (sec). Leaf size: 87

ode=D[y[x],{x,2}]+Tan[x]*D[y[x],x]==Sin[2*x]; 
ic={y[0]==-1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\cos (K[2]) \left (\int _1^{K[2]}2 \sin (K[1])dK[1]-\int _1^02 \sin (K[1])dK[1]\right )dK[2]-\int _1^0\cos (K[2]) \left (\int _1^{K[2]}2 \sin (K[1])dK[1]-\int _1^02 \sin (K[1])dK[1]\right )dK[2]-1 \end{align*}

✓ Sympy. Time used: 0.546 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(2*x) + tan(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x \sin ^{2}{\left (x \right )} - x \cos ^{2}{\left (x \right )} - \sin {\left (x \right )} \cos {\left (x \right )} + 2 \sin {\left (x \right )} - 1 \]