1. The problem involves understanding and solving various types of differential equations including exact differential equations, linear differential equations, and growth/decay problems. 2. For exact differential equations, the general form is $$M(x,y) + N(x,y)\frac{dy}{dx} = 0$$ where $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$ ensures exactness. 3. To solve, find a potential function $$\psi(x,y)$$ such that $$\frac{\partial \psi}{\partial x} = M$$ and $$\frac{\partial \psi}{\partial y} = N$$. Then the solution is $$\psi(x,y) = C$$. 4. For linear differential equations of first order, the form is $$\frac{dy}{dx} + P(x)y = Q(x)$$. Use the integrating factor $$\mu(x) = e^{\int P(x) dx}$$ to solve. 5. Growth and decay problems follow $$\frac{dy}{dt} = ky$$ with solution $$y = y_0 e^{kt}$$ where $$k$$ is the growth/decay constant. 6. Newton's Law of Cooling is $$\frac{dT}{dt} = -k(T - T_{env})$$ with solution $$T = T_{env} + (T_0 - T_{env})e^{-kt}$$. 7. Mixing problems involve setting up differential equations based on rates of inflow and outflow of substances. This overview covers the formation and solution methods for the mentioned differential equations and applications.