In each of the following problems use one of the substitutions discussed in this chapter to find a general solution to thedifferential equation.
\[\frac{dy}{dt} = \frac{1}{y+t}\]
\[\frac{dy}{dt} = (4t+9y)^2\]
\[y' = \frac{1}{\sqrt{y+t}}\]
\[\dot{y} = e^{2t+y}\]
\[\frac{dy}{dt} = \tan^2(y+t)\]
\[y' = \frac{y\ln(y/x^2)}{x}\]
\[\frac{dy}{dx} = \frac{e^{xy}}{x^2} - \frac{y}{x}\]
(HINT: Let \(z=xy\))
\[y' = t^2\cos(y/t^3)+3y/t\]
\[\frac{dy}{dx} = \frac{\sqrt{x^2 + y^2}}{x}\]
Solve each of the following Bernoulli equations using (a) Leibniz’s substitution and (b) Bernoulli’s substitution.
\[y' = 2y+ty^2\]
\[y' = y + t^2/y\]
\[y' = \frac{t-y^3}{ty^2}\]