linear differential equation,
n. 1. an ordinary differential equation that does not contain any products (including powers greater than 1) of the derivatives and the dependent variable. A first order linear ordinary differential equation has the form
which is homogeneous if b(x) = 0. There is an algebraic technique for the solution of homogeneous linear equations, which is always effective if all the coefficients in the equation are constants and the order is less than 5. A second-order linear differential equation can be reduced to a first-order linear equation if the differential operator can be factorized into linear factors, or if a particular solution of the homogeneous equation is known. If the complementary function is known, the general solution can be found by variation of parameters. See Euler equation, Frobenius method. 2. a partial differential equation that does not contain any products (including powers greater than 1) of partial derivatives and the dependent variable. A complete solution of such an equation can be found as the sum of a complementary function, which is a complete solution of the homogeneous equation, and a particular integral. There is an algebraic technique, similar to the one for ordinary differential equations, for finding a complete solution of an equation of the form
where x and y are independent variables, ai is a constant, and f(x, y) is a differentiable function. See Lagrange's linear equation.
y' + a(x) y = b(x),
which is homogeneous if b(x) = 0. There is an algebraic technique for the solution of homogeneous linear equations, which is always effective if all the coefficients in the equation are constants and the order is less than 5. A second-order linear differential equation can be reduced to a first-order linear equation if the differential operator can be factorized into linear factors, or if a particular solution of the homogeneous equation is known. If the complementary function is known, the general solution can be found by variation of parameters. See Euler equation, Frobenius method. 2. a partial differential equation that does not contain any products (including powers greater than 1) of partial derivatives and the dependent variable. A complete solution of such an equation can be found as the sum of a complementary function, which is a complete solution of the homogeneous equation, and a particular integral. There is an algebraic technique, similar to the one for ordinary differential equations, for finding a complete solution of an equation of the form