Differential Equations: First Order Linear and Separable Equations

Using substitution methods, solve a first-order differential equation that seems non-separable at first.

Solve the differential equation $\frac{du}{dt} = 2t + \frac{\sec^2 t}{2u}$ with the initial condition $u(0) = -5$.

Solve the differential equation $\frac{dy}{dx} - 3y = 0$ with the initial condition $y(0) = -1$.

Solve the differential equation $y' = \frac{x^2}{y}$ by showing that it is separable and using integration to find the solution.

Solve the initial value problem: $y' = 8xy + 3y$ with $y(0) = 5$.

Find the values of the constants $m$ and $b$ for which $y = mx + b$ is a solution to the differential equation $\frac{dy}{dx} = \frac{1}{2}x + y - 1$.

If I'm looking at the point (1, -2), write the equation for the line tangent to the function that satisfies the differential equation through this point.