By Sapagovas M.P.

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**Extra info for A Boundary Value Problem with a Nonlocal Condition for a System of Ordinary Differential Equations**

**Example text**

0 by separation of variables we have P(x)dx+A So the general solution is Y= eelP(x)dx = Yc (b) We have yp = eJ P(x)dx ! I(X)Q(X) dx + eJ P(x)dXI(x)Q(x) dx =P(x)yP + Q(x) Thus, YP is a particular solution of the inhomogeneous equation. The results of (a) and (b) foreshadow a powerful theorem of linear equations in general, which occurs repeatedly in different contexts throughout mathematics: • Theorem - - - - - - - - - - - - - - - - - - The general solution of an inhomogeneous linear first-order differential equation may be written as the sum of the general solution of the corresponding homogeneous equation (putting Q(x) == 0) - called the complementary function - and a particular solution of the inhomogeneous equation - called a particular integral.

It can then be shown that 46 Ordinary Differential Equations the sequence Yo(x), Yl(x), ... ,Yr(X), . converges to an exact solution if the conditions of the Peano existence theorem are satisfied. In practice, this method is a poor way to obtain a solution - its main purpose is a theoretical one, used in establishing existence and uniqueness theorems. 6 Obtain the first four Picard approximations to the solutions of the initial value problem y' = 1 + y 2 , yeO) = 0, and compare with the exact answer.

The results of (a) and (b) foreshadow a powerful theorem of linear equations in general, which occurs repeatedly in different contexts throughout mathematics: • Theorem - - - - - - - - - - - - - - - - - - The general solution of an inhomogeneous linear first-order differential equation may be written as the sum of the general solution of the corresponding homogeneous equation (putting Q(x) == 0) - called the complementary function - and a particular solution of the inhomogeneous equation - called a particular integral.

### A Boundary Value Problem with a Nonlocal Condition for a System of Ordinary Differential Equations by Sapagovas M.P.

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