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 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?
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 $[0,1]$.