73.23.34 problem 33.11 (h)

Internal problem ID [15680]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.11 (h)
Date solved : Tuesday, January 28, 2025 at 08:04:05 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }+\cos \left (y\right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 59

Order:=5; 
dsolve(diff(y(x),x)+cos(y(x))=0,y(x),type='series',x=0);
 

\[ y = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {\sin \left (2 y \left (0\right )\right ) x^{2}}{4}+\frac {\cos \left (y \left (0\right )\right ) \cos \left (2 y \left (0\right )\right ) x^{3}}{6}+\left (\frac {\sin \left (4 y \left (0\right )\right )}{32}+\frac {\sin \left (2 y \left (0\right )\right )}{24}\right ) x^{4}+O\left (x^{5}\right ) \]

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 194

AsymptoticDSolveValue[D[y[x],x]+Cos[y[x]]==0,y[x],{x,0,"5"-1}]
 

\[ y(x)\to 2 \arctan \left (\tanh \left (\frac {c_1}{2}\right )\right )+\frac {x^4 \left (-5 \tanh ^7\left (\frac {c_1}{2}\right )+19 \tanh ^5\left (\frac {c_1}{2}\right )-19 \tanh ^3\left (\frac {c_1}{2}\right )+5 \tanh \left (\frac {c_1}{2}\right )\right )}{12 \left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^4}+\frac {x^3 \left (1-\tanh ^6\left (\frac {c_1}{2}\right )+7 \tanh ^4\left (\frac {c_1}{2}\right )-7 \tanh ^2\left (\frac {c_1}{2}\right )\right )}{6 \left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^3}+\frac {x^2 \left (\tanh ^3\left (\frac {c_1}{2}\right )-\tanh \left (\frac {c_1}{2}\right )\right )}{\left (1+\tanh ^2\left (\frac {c_1}{2}\right )\right ){}^2}+\frac {x \left (-1+\tanh ^2\left (\frac {c_1}{2}\right )\right )}{1+\tanh ^2\left (\frac {c_1}{2}\right )} \]