Internal problem ID [2203]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\left (\sec ^{2}\relax (y)\right ) y^{\prime }+\frac {\tan \relax (y)}{2 \sqrt {x +1}}-\frac {1}{2 \sqrt {x +1}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \relax (x ) = \arctan \left ({\mathrm e}^{-\sqrt {x +1}} c_{1}+1\right ) \]
✓ Solution by Mathematica
Time used: 38.437 (sec). Leaf size: 248
DSolve[Sec[y[x]]^2*y'[x]+1/(2*Sqrt[1+x])*Tan[y[x]]==1/(2*Sqrt[1+x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\text {ArcCos}\left (-\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {1+2 e^{\sqrt {x+1}+2 c_1} \left (-1+e^{\sqrt {x+1}+2 c_1}\right )}}\right ) \\ y(x)\to \text {ArcCos}\left (-\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {1+2 e^{\sqrt {x+1}+2 c_1} \left (-1+e^{\sqrt {x+1}+2 c_1}\right )}}\right ) \\ y(x)\to -\text {ArcCos}\left (\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {1+2 e^{\sqrt {x+1}+2 c_1} \left (-1+e^{\sqrt {x+1}+2 c_1}\right )}}\right ) \\ y(x)\to \text {ArcCos}\left (\frac {e^{\sqrt {x+1}+2 c_1}}{\sqrt {1+2 e^{\sqrt {x+1}+2 c_1} \left (-1+e^{\sqrt {x+1}+2 c_1}\right )}}\right ) \\ y(x)\to \frac {\pi }{4} \\ \end{align*}