Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS f(x)=0: an(x) dny dxn +an− 1 (x) dn−^1 y dxn−^1 +···+a 1 (x) dy dx +a 0 (x)y=0. (15 ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS If the original equation (15.1) hasf(x) = 0 (i.e. it is homogeneous) then of course ...

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS In general the auxiliary equation hasnroots, sayλ 1 ,λ 2 ,...,λn. In certain ca ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Find the complementary function of the equation d^2 y dx^2 − 2 dy dx +y=ex. (15.15 ...

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS (iv) Iff(x) is the sum or product of any of the above then tryyp(x)asthe sum or ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Solve d^2 y dx^2 +4y=x^2 sin 2x. (15.16) First we set the RHS to zero and assume t ...

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS constant coefficients and possess simple functions on the RHS. Such equations o ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS and, ifkis a constant, the particular solution is equally straightforward:wn=K for ...

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS that a particular solution of the formun=Aαnshould be tried. Substituting this ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS both linear and homogeneous, and is satisfied by bothvn=Aλn 1 andvn=Bλn 2 ,its gene ...

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS Higher-order recurrence relations It will be apparent that linear recurrence re ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Solve d^2 y dx^2 − 3 dy dx +2y=2e−x, (15.33) subject to the boundary conditionsy(0 ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS Using table 13.1, q 1 (t)=^12 V 0 C(cosω 1 t−cosω 2 t), whereω^21 (L+M)=Gandω 2 ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS Substituting equations (15.37) into the original equation (15.36), the latter becom ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS y=xλ. This leads to an algebraic equation whose solution gives the allowed valu ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS It is worth noting that, even if a higher-order ODE is not exact in its given form, ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
Solve d^2 y dx^2 +y=cosecx. (15.50) We see that the RHS does not fall into any ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS 15.2.4 Variation of parameters The method of variation of parameters proves useful ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS we are free to choose our constraints as we wish, let us define the expression ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS integration equal to zero, to givekm(x). The general solution to (15.53) is then gi ...