Can anyone see why these integrals are necessarily equal?

Is it possible find a function $u(x)$ so that $[y'(x)+y(x)\tan(x)]^2=(u'(x))^2-(u(x))^2$?

If not, is there an obvious reason why the integrals of the LHS an the RHS respectively over the interval $(0,1)$ are equal?

• Is this of any help? $$u(x) = y(x) \sec(x)\\ u'(x) = y'(x) \sec(x) + y(x) \sec(x) \tan(x)$$ $$\left(u'(x) \right)^2 - (u(x))^2= (y'(x) + y(x) \tan(x))^2 \sec^2(x) - y^2(x) \sec^2(x)$$ – user17762 May 23 '12 at 21:05
• @Marvis: Thanks for the suggestion! – freda johnson May 23 '12 at 21:27
• but at the moment i am still not entirely sure of how this works. maybe integrating by parts would help? – freda johnson May 23 '12 at 21:31
• I don't understand your second question. The integrals of what over that interval are equal? – Greg Martin May 23 '12 at 22:17
• @GregMartin: the LHS and the RHS respectively. – freda johnson May 23 '12 at 22:26

To get a suitable $u$, you might solve the differential equation $u'(x) = \sqrt{(y'(x)+y(x) \tan(x))^2 + u(x)^2}$. Assuming $y$ and $y'$ are continuous on $[0,1]$, the right side of that differential equation is continuous and locally Lipschitz on $[0,1] \times \mathbb R$, so for any initial condition at some $x_0 \in [0,1]$ we have local existence and uniqueness of solutions. Moreover, since $\sqrt{A^2 + u^2} \le |A| + |u|$, the solution will exist on all of $[0,1]$.