Identify and describe basic mathematical models using differential equations.
Analyze and interpret direction fields for visualizing solutions of first-order differential equations.
Solve basic first-order differential equations analytically.
Verify that specific functions are solutions to given differential equations.
Solve linear equations using the method of integrating factors.
Solve separable differential equations.
Develop and analyze mathematical models using ODEs.
Analyze autonomous differential equations and apply them to population dynamics models.
Determine if a differential equation is exact and solve it using integrating factors if necessary.
Apply Euler’s method for numerical approximation of solutions to first-order differential equations.
Understand and explain the existence and uniqueness theorem for first-order differential equations.
Solve homogeneous linear equations with constant coefficients.
Understand the concept of the Wronskian and use it to determine the linear independence of solutions.
Solve linear differential equations using the characteristic equation.
Solve nonhomogeneous linear equations using the method of undetermined coefficients.
Apply the method of variation of parameters to solve nonhomogeneous linear equations.
Apply to mechanical and electrical vibrations.
Find series solutions for second-order linear equations.
Understand the general theory and structure of nth order linear differential equations.
Solve nth order homogeneous linear differential equations with constant coefficients.
Extend the method of undetermined coefficients to solve higher-order nonhomogeneous differential equations.
Apply the method of variation of parameters to higher-order differential equations.
Apply to physical problems.
Define and compute the Laplace transform of functions.
Apply the Laplace transform to solve initial value problems involving linear differential equations.
Solve ODEs with discontinuous forcing functions.
Use the convolution theorem to solve integro-differential equations.
Understand the basic theory underlying systems of first-order linear differential equations.
Solve homogeneous systems of first-order linear differential equations with constant coefficients.
Solve nonhomogeneous systems of first-order linear equations.
Apply to physical problems.
Apply the Euler method to approximate solutions of differential equations.
Understand and apply improvements to the Euler method for better accuracy in numerical solutions.
Implement the Runge-Kutta method to numerically solve differential equations.