83.5.20 problem 20

Internal problem ID [19115]
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 (D) at page 16
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 12:58:05 PM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 78

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

\begin{align*} y \left (x \right ) &= \frac {\sqrt {-\cos \left (x \right ) \left (\cos \left (x \right )+\left (-1+\sin \left (x \right )\right ) \left (x +c_{1} \right )\right )}}{\left (x +c_{1} \right ) \sin \left (x \right )+\cos \left (x \right )-c_{1} -x} \\ y \left (x \right ) &= -\frac {\sqrt {-\cos \left (x \right ) \left (\cos \left (x \right )+\left (-1+\sin \left (x \right )\right ) \left (x +c_{1} \right )\right )}}{\left (x +c_{1} \right ) \sin \left (x \right )+\cos \left (x \right )-c_{1} -x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.448 (sec). Leaf size: 136

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

\begin{align*} y(x)\to -\frac {i e^{\text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} \sqrt {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}}{\sqrt {(x+2+c_1) \sin \left (\frac {x}{2}\right )-(x+c_1) \cos \left (\frac {x}{2}\right )}} \\ y(x)\to \frac {i e^{\text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} \sqrt {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}}{\sqrt {(x+2+c_1) \sin \left (\frac {x}{2}\right )-(x+c_1) \cos \left (\frac {x}{2}\right )}} \\ y(x)\to 0 \\ \end{align*}