Practice section 1 of 6

Find the general solution of $dy/dx=2x/(x^2+1)$.

(a) Solve $\cos^2x\,dy/dx=1$ for $0<x<\pi/2$, given $y=0$ when $x=\pi/4$, expressing y as a function of x.

(b) Find the equation of a curve through $(1,2)$ and $(1/4,-10)$ whose gradient is inversely proportional to $x^2$.

Find the general solution of $dy/dx=2x+\sin x+5$.

Find the general solution of $dy/dx-2y=0$.

Solve the equation from Section 1, $dN/dt=-kN$, where $k>0$.

Solve $dy/dx=x^2/y$, given $y=1$ when $x=1$.