15.8.30 problem 31

Internal problem ID [3033]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 31
Date solved : Monday, January 27, 2025 at 07:09:55 AM
CAS classification : [_separable]

\begin{align*} 3 \,{\mathrm e}^{x} \tan \left (y\right )&=\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 144

dsolve(3*exp(x)*tan(y(x))=(1-exp(x))*sec(y(x))^2*diff(y(x),x),y(x), singsol=all)
 

\[ y = \frac {\arctan \left (\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}, \frac {{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}-c_{1}^{2}-6 \,{\mathrm e}^{x}+1}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}\right )}{2} \]

✓ Solution by Mathematica

Time used: 1.295 (sec). Leaf size: 78

DSolve[3*Exp[x]*Tan[y[x]]==(1-Exp[x])*Sec[y[x]]^2*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\right )+2 c_1\right )\right ) \\ y(x)\to \frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\right )+2 c_1\right )\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}