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?
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The highest derivative present in the equation.- What is the degree of a differential equation?
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The power to which the highest order derivative is raised.- Define an ordinary differential equation (ODE).
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An equation involving derivatives with respect to a single independent variable.- Define a partial differential equation (PDE).
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An equation involving partial derivatives with respect to multiple independent variables.- What is a linear differential equation?
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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?
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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?
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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?
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The complete family of all solutions, typically containing one or more arbitrary constants.- What is a particular solution to a differential equation?
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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?
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Separate variables by writing (dy/h(y)) = g(x)dx, then integrate both sides.- What is an integrating factor?
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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?
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mu(x) = e^(integral of P(x)dx)- What is an exact differential equation?
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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?
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A determinant W = y1(y2') - y2(y1') used to test linear independence of two solutions y1 and y2.- What is linear independence of solutions?
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Two solutions y1 and y2 are linearly independent if one is not a constant multiple of the other (equivalently, their Wronskian is non-zero).
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