75.4.26 problem 91

Internal problem ID [16728]
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 : Tuesday, January 28, 2025 at 09:19:42 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 18

dsolve(tan(diff(y(x),x))=x,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[Tan[D[y[x],x]]==x,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*}