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 : Sunday, March 30, 2025 at 02:05:13 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*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 17

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

\[ y = \arctan \left ({\mathrm e}^{-\sqrt {1+x}} c_1 +1\right ) \]

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

ode=Sec[y[x]]^2*D[y[x],x]+1/(2*Sqrt[1+x])*Tan[y[x]]==1/(2*Sqrt[1+x]); 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 2.354 (sec). Leaf size: 134

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x)/cos(y(x))**2 + tan(y(x))/(2*sqrt(x + 1)) - 1/(2*sqrt(x + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \operatorname {atan}{\left (\frac {\sqrt {2 \sqrt {2} + 3} \left (e^{\sqrt {2 \sqrt {2} + 3} \left (2 C_{1} + \sqrt {2} \sqrt {x + 1}\right )} + e^{\sqrt {2 \sqrt {2} + 3} \left (2 \sqrt {2} C_{1} + \sqrt {x + 1}\right )}\right )}{e^{\sqrt {2 \sqrt {2} + 3} \left (2 C_{1} + \sqrt {2} \sqrt {x + 1}\right )} - e^{\sqrt {2 \sqrt {2} + 3} \left (2 \sqrt {2} C_{1} + \sqrt {x + 1}\right )}} \right )} - \frac {\pi }{8} \]