Internal problem ID [11739]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 18, The variation of constants formula. Exercises page 168
Problem number: 18.1 (vi).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {\left (\tan \left (x \right )^{2}-1\right ) y^{\prime \prime }-4 \tan \left (x \right )^{3} y^{\prime }+2 y \sec \left (x \right )^{4}=\left (\tan \left (x \right )^{2}-1\right ) \left (1-2 \sin \left (x \right )^{2}\right )} \] Given that one solution of the ode is \begin {align*} y_1 &= \sec \left (x \right )^{2} \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 29
dsolve([(tan(x)^2-1)*diff(y(x),x$2)-4*tan(x)^3*diff(y(x),x)+2*y(x)*sec(x)^4=(tan(x)^2-1)*(1-2*sin(x)^2),sec(x)^2],y(x), singsol=all)
\[ y \left (x \right ) = \sec \left (x \right )^{2} c_{2} +\sec \left (x \right ) \sin \left (x \right ) c_{1} -\frac {\cos \left (x \right )^{2}}{4}+\frac {x \tan \left (x \right )}{2}+\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 0.764 (sec). Leaf size: 66
DSolve[(Tan[x]^2-1)*y''[x]-4*Tan[x]^3*y'[x]+2*y[x]*Sec[x]^4==(Tan[x]^2-1)*(1-2*Sin[x]^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {\sin ^2(x)} \sec (x) \arctan \left (\frac {\cos (x)}{1-\sqrt {\sin ^2(x)}}\right )-\frac {1}{4} \cos ^2(x)+c_1 \sec ^2(x)+c_2 \sqrt {\sin ^2(x)} \sec (x)+\frac {1}{2} \]