58.2.25 problem 25

Internal problem ID [9148]
Book : Second order enumerated odes
Section : section 2
Problem number : 25
Date solved : Monday, January 27, 2025 at 05:49:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \cos \left (x \right ) y^{\prime \prime }+\sin \left (x \right ) y^{\prime }-2 y \cos \left (x \right )^{3}&=2 \cos \left (x \right )^{5} \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 30

dsolve(cos(x)*diff(y(x),x$2)+sin(x)*diff(y(x),x)-2*y(x)*cos(x)^3=2*cos(x)^5,y(x), singsol=all)
 

\[ y = \sinh \left (\sin \left (x \right ) \sqrt {2}\right ) c_{2} +\cosh \left (\sin \left (x \right ) \sqrt {2}\right ) c_{1} +\frac {1}{2}-\frac {\cos \left (2 x \right )}{2} \]

✓ Solution by Mathematica

Time used: 11.572 (sec). Leaf size: 90

DSolve[Cos[x]*D[y[x],{x,2}]+Sin[x]*D[y[x],x]-2*y[x]*Cos[x]^3==2*Cos[x]^5,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-i \sqrt {\cos (2 x)-1}} \left (\cos \left (\sqrt {\cos (2 x)-1}\right )+i \sin \left (\sqrt {\cos (2 x)-1}\right )\right ) \left (-\cos (2 x)+2 c_1 \cos \left (\sqrt {\cos (2 x)-1}\right )+2 c_2 \sin \left (\sqrt {\cos (2 x)-1}\right )+1\right ) \]