First and second order differential equation solution methods and classification concepts from an introductory college course.
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- What is the order of a differential equation?
- The highest derivative present in the equation.
- What is the degree of a differential equation?
- The power to which the highest order derivative is raised.
- Define an ordinary differential equation (ODE).
- An equation involving derivatives with respect to a single independent variable.
- Define a partial differential equation (PDE).
- An equation involving partial derivatives with respect to multiple independent variables.
- What is a linear differential equation?
- An equation where the dependent variable and its derivatives appear only to the first power, with no products of these terms.
- What is a homogeneous first order ODE?
- An equation that can be written as dy/dx = f(y/x), where the right side depends only on the ratio y/x.
- What is a separable differential equation?
- An equation that can be written as dy/dx = g(x)h(y), separating variables on each side.
- What is the general solution to a differential equation?
- The complete family of all solutions, typically containing one or more arbitrary constants.
- What is a particular solution to a differential equation?
- A specific solution obtained by assigning particular values to the arbitrary constants in the general solution, often using initial conditions.
- To solve a separable equation dy/dx = g(x)h(y), what is the first step?
- Separate variables by writing (dy/h(y)) = g(x)dx, then integrate both sides.
- What is an integrating factor?
- A function mu(x) used to multiply a linear ODE so it becomes exact and solvable.
- For a linear first order ODE y' + P(x)y = Q(x), what is the standard integrating factor?
- mu(x) = e^(integral of P(x)dx)
- What is an exact differential equation?
- An equation M(x,y)dx + N(x,y)dy = 0 where dM/dy = dN/dx, so it equals dF = 0 for some function F.
- What is the Wronskian?
- A determinant W = y1(y2') - y2(y1') used to test linear independence of two solutions y1 and y2.
- What is linear independence of solutions?
- Two solutions y1 and y2 are linearly independent if one is not a constant multiple of the other (equivalently, their Wronskian is non-zero).