Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 16 / b

Near the initial point, division by $\cos x$ gives $y^{\prime }+(1+\tan x)y=2\sec x+2\sin x$. An integrating factor is $e^x\sec x$, andThus $y=2\sin x+C{e}^{-x}\cos x$, and $y(0)=2$ givesAlthough division by $\cos x$ was local, direct substitution verifies this smooth solution in the original equation for every real $x$, including the zeros of $\cos x$.

Set $u=1/y$, so $u^{\prime }=-y^{\prime }/y^2$. The equation becomes $x(x^2-1)u^{\prime }+u=-x^3$. For $x>1$ its integrating factor is $\sqrt{x^2-1}/x$, givingThe initial value is $u(2)=-2$, forcing $C=0$. HenceSubstitution into the original equation verifies this answer throughout $x>0$, including $x=1$ where the divided linear equation was singular. Its maximal connected interval through the initial point is $(0,\infty )$.