Internal problem ID [13226]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.2 (k).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sin \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1+\cos \left (x \right )^{2}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve([sin(x)^2*diff(y(x),x$2)-2*cos(x)*sin(x)*diff(y(x),x)+(1+cos(x)^2)*y(x)=0,sin(x)],y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sin \left (x \right )+c_{2} \sin \left (x \right ) x \]
✓ Solution by Mathematica
Time used: 0.079 (sec). Leaf size: 15
DSolve[Sin[x]^2*y''[x]-2*Cos[x]*Sin[x]*y'[x]+(1+Cos[x]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) \sin (x) \]