75.20.6 problem 641

Internal problem ID [17140]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 641
Date solved : Tuesday, January 28, 2025 at 09:54:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\cos \left (\sin \left (x \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.352 (sec). Leaf size: 15

dsolve([diff(y(x),x$2)+tan(x)*diff(y(x),x)+cos(x)^2*y(x)=0,cos(sin(x))],singsol=all)
 

\[ y = c_{1} \sin \left (\sin \left (x \right )\right )+c_{2} \cos \left (\sin \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 1.898 (sec). Leaf size: 37

DSolve[D[y[x],{x,2}]+Tan[x]*D[y[x],x]+Cos[x]^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cosh \left (\sqrt {-\sin ^2(x)}\right )+i c_2 \sinh \left (\sqrt {-\sin ^2(x)}\right ) \]