# Linear Differential Equations

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## Linear Differential Equations

differential equations of the form

(1) *y*^{(n)} + *p*_{1}(*x*)*y*^{(n - 1)} + ... + *p*_{n}(*x*)*y* = *f*(*x*)

where *y* = *y*(*x*) is an unknown function; *y*^{(n)}, *y*^{(n - 1)}, . . ., *y* are its derivatives; and *p*_{1}(*x*), *p*_{2}(*x*), . . ., *p*_{n} (*x*) (the coefficients) and/(x) (the free term) are given functions. Equation (1) is said to be linear because the unknown function *y* and its derivatives enter (1) linearly, that is, the degree of *y* and its derivatives in (1) is one. If *f(x)* ≡ 0, then equation (1) is said to be homogeneous; otherwise it is inhomogeneous. The general solution *y*_{0} = *y*_{0}(*x*) of a homogeneous linear differential equation with continuous coefficients *p*_{k}(*x*) can be written as

*y*_{0} = *C*_{1}*y*_{1}(*x*) + *C*_{2}*y*_{2}(*x*) + ... + *C*_{n}*y*_{n}(*x*)

where *C*_{1}, *C*_{2}, . . ., *C*_{n} are arbitrary constants and *y*_{1}(*x*), *y*_{2}(*x*), . . ., *y*_{n}(*x*) are linearly independent particular solutions. Such *n* solutions are said to form a fundamental system of solutions. An important result states that *n* particular solutions are linearly independent if and only if their Wronskian

is different from zero at at least one point.

The general solution *y* = *y*(*x*) of the inhomogeneous linear differential equation (1) has the form

*y* = *y*_{0} + *Y*

where *y*_{0} = *y*_{0}(*x*) is the general solution of the corresponding homogeneous linear differential equation and *Y* = *Y*(*x*) is a particular solution of the given inhomogeneous linear differential equation. The function *Y*(*x*) can be found from the formula

where the *y*_{k}(*x*) form a fundamental system of solutions of the homogeneous linear differential equation and *W*_{k}(*x*) is the cofactor of the element *y*_{k}^{(n - 1)}(*x*) in the Wronskian *w*(*x*) given above in (2).

If the coefficients of equation (1) are constant, that is, *p*_{k}(*x*) = *a*_{k} for *k* = 1, 2, . . ., *n*, then the general solution of the homogeneous equation is given by

Here, ) are the roots of the so-called characteristic equation

λ^{n} + *a*_{1} λ^{n - 1} + ... + *a*_{n} = 0

and the *n*_{k} are the multiplicities of these roots and *C*_{ks} and *D*_{ks} are arbitrary constants.

*Example.* For the linear differential equation *y*″′ + *y* = 0, the characteristic equation has the form λ^{3} + 1 = 0. Its roots are

Consequently, the general solution of this equation is

Consider a system of *n* linear differential equations

The general solution of the homogeneous system of linear differential equations [obtained from (3) by putting all *f*_{j}(*x*) ≡ 0] is given by

where *y*_{j1}, *y*_{j 2}, . . ., *y*_{jn} are linearly independent particular solutions of the homogeneous system (that is, solutions such that the determinant ǀ*y*_{jk} (x)ǀ≠ 0 at at least one point).

In the case of constant coefficients *p*_{jk}(*x*) = *a*_{jk}, particular solutions of the homogeneous system are of the form

where the *A*_{js} are undetermined coefficients and the λ_{k} are the roots of the characteristic equation

with multipliers *m*_{k}. Here, a complete analysis of all possible cases can be effected by means of the theory of elementary divisors.

The methods of operational calculus are also used to solve linear differential equations and systems of linear differential equations with constant coefficients.

### REFERENCES

Stepanov, V. V.*Kurs differentsial’nykh uravnenii,*8th ed. Moscow, 1959.

Smirnov, V. I.

*Kurs vysshei matematiki.*Vol. 2, 20th ed., Moscow, 1967; vol. 3, part 2, 8th ed., Moscow, 1969.

Pontriagin, L. S.

*Obyknovennye differentsial’nye uravneniia,*3rd ed. Moscow, 1970.