83.3.15 problem 15

Internal problem ID [19071]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (B) at page 9
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 12:51:49 PM
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 }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 202

dsolve((3*exp(x)*tan(y(x)))+(1-exp(x))*sec(y(x))^2*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.013 (sec). Leaf size: 74

DSolve[(3*Exp[x]*Tan[y[x]])+(1-Exp[x])*Sec[y[x]]^2*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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