20.4.48 problem Problem 67

Internal problem ID [3683]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 67
Date solved : Monday, January 27, 2025 at 07:54:47 AM
CAS classification : [_separable]

\begin{align*} \sec \left (y\right )^{2} y^{\prime }+\frac {\tan \left (y\right )}{2 \sqrt {1+x}}&=\frac {1}{2 \sqrt {1+x}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 17

dsolve(sec(y(x))^2*diff(y(x),x)+1/(2*sqrt(1+x))*tan(y(x))=1/(2*sqrt(1+x)),y(x), singsol=all)
 

\[ y \left (x \right ) = \arctan \left ({\mathrm e}^{-\sqrt {x +1}} c_{1} +1\right ) \]

✓ Solution by Mathematica

Time used: 60.256 (sec). Leaf size: 247

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

\begin{align*} y(x)\to -\arccos \left (-\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {-2 e^{\sqrt {x+1}+2 c_1}+2 e^{2 \sqrt {x+1}+4 c_1}+1}}\right ) \\ y(x)\to \arccos \left (-\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {-2 e^{\sqrt {x+1}+2 c_1}+2 e^{2 \sqrt {x+1}+4 c_1}+1}}\right ) \\ y(x)\to -\arccos \left (\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {-2 e^{\sqrt {x+1}+2 c_1}+2 e^{2 \sqrt {x+1}+4 c_1}+1}}\right ) \\ y(x)\to \arccos \left (\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {-2 e^{\sqrt {x+1}+2 c_1}+2 e^{2 \sqrt {x+1}+4 c_1}+1}}\right ) \\ \end{align*}