14. starptautiskā konference 2012 - Latvijas Jūras akadēmija

Proceedings of 14th International conference „Maritime Transport and Infrastructure - 2012”2 2cy C1cost C2sint cosct(11)2 2 Now it is allowed to determine undetermined coefficients of integration C1un C2. If then the equivalence j , t.i., j j is accomplished. cAccession:ccy te0ta t b sin t tacos t b sin t0cosc 0 c 0 c 0c. (12)Y function in equation (1) is replaced by function (12). Two differential equations areacquired for solving coefficients a0un b0by comparing coefficients of equal functions. If then a particular solution (12) is acquired. cy 3 t sin ct2Also the general solution (2) of equation (1) is acquired:(13)3y C1 cost C2sint t sinct. (14)2Now it is allowed to determine undetermined integration constants C 1andC2.The constants variation method of LagrangeThe constant variant method is more general than method which was mentioned earlier. It ispossible to use it for solving any linear differential equation [3,4] independently from the type of theright side of equation but the method is connected with solution of integrals. The solution of theequation can be left in integral form therefore the solution of differential equation sometimes is calledas an integral of equation.Let us check the equation (1) whose right side is undetermined function f(t).t y y f . (15)2 The solution of appropriate homogenous equation (3) should be stated using formula:where t un y ty2tty C1 y1C2y2(16)1are undetermined, linearly independent solutions of homogenousequation. Now it is necessary to change (variate) integrating constants so that function (16) couldsatisfy non – homogenous equation (15), t.s, accept that constants are dependent from t:tC1 C1un C2 C2(17)The solutions must be written in form:ttyt C tyty C1 12 2. (18)81