75.4.26 problem 91

Internal problem ID [16657]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 91
Date solved : Monday, March 31, 2025 at 03:03:49 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

ode:=tan(diff(y(x),x)) = x; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 163

ode=Tan[D[y[x],x]]==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -x \arccos \left (-\frac {1}{\sqrt {x^2+1}}\right )-\frac {\sqrt {x^2} \log \left (x^2+1\right )}{2 x}+c_1 \\ y(x)\to x \arccos \left (\frac {1}{\sqrt {x^2+1}}\right )-\frac {\sqrt {x^2} \log \left (x^2+1\right )}{2 x}+c_1 \\ y(x)\to x \arccos \left (-\frac {1}{\sqrt {x^2+1}}\right )+\frac {\sqrt {x^2} \log \left (x^2+1\right )}{2 x}+c_1 \\ y(x)\to -x \arccos \left (\frac {1}{\sqrt {x^2+1}}\right )+\frac {\sqrt {x^2} \log \left (x^2+1\right )}{2 x}+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} + x \operatorname {atan}{\left (x \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \]