Differential Equations for Calculus II
1. Methods of solving differential equations.
• Separation of Variable
• Integrating factor
• Know solution to dy/dx=ay and dy/dx=ay, y(0)=c
• Know Maple syntax for solving differential equations
2. Exponential Growth/Decay problems
• Set up for continuous compounding problems (i.e. start from A(t+∆t)-A(t) and
end with the differential equation dA/dt=rA.
• Using separation of variables to find the general solution dy/dx=ay
• Know at any given time that the solution of dy/dx=αy is y=ceαx .
• Know how to use the exponential equation to solve continuous compounding,
bacteria growth, and radio active decay problem
3. Newton’s Law of cooling
• Know the statement leading to the differential equation.
• Know how to use separation of variables to solve dT/dt = α(T(t)-Ts).
4. Salt tank problems (one tank)
• Rate in = rate out (Can use separation of variable)
• Rate in ≠ rate out (Can not use separation of variable – will need the integrating
factor technique)
5. Falling Body problems.
• Assume that gravity is the only force (easy integration problem)
• Assume that wind resistance is also a force on the object (equation is now a
differential equation).
• Note: we will work hard on understanding how to set these problems up with
different coordinate systems.
6. Interest rate problems
• Continuous compounding
• Depositing or withdrawing continuously
7. Direction Fields
• Simple ones by hand
• Using Maple
• Equilibrium solutions
• Stability of equilibrium solutions
8. Population problems
• Use the direction field to analyze
• Solve with Maple