Find the differential of each function. (a) \(y = \tan \sqrt{t}\) (b) \(y= \frac{1-v^2}{1+v^2}\)

If \(f(x) + x^2[f(x)]^5 = 34\) and \(f(1) = 2,\) find \(f '(1).\)

Solve differential equation \(y '(t) = -3y + 9\), \(y(0) = 4\)

Solve differential equation \(\frac{dy}{dx}+(\frac{a}{x})y=40x\), for \(x>0\) and \(y(1)=a\)

Solve the differential equation \(y'-\lambda y= 1-\lambda t\), \(y(0)=0\)

Solve differential equation \((1+y^2+xy^2)dx+(x^2y+y+2xy)dy=0\)

Solve differential equation \(\frac{\cos^2y}{4x+2}dy= \frac{(\cos y+\sin y)^2}{\sqrt{x^2+x+3}}dx\)

Solve differential equation \(u'-5u=ve^{-5v}\)