Second order linear differential equations - Douis.net
iated with this equation is. The auxiliary equation is a quadratic equation, three cases are possible: 0 has two distinct solutions. The complemen. 0 tary function i.
d2y dy b cy 0. 2 dx dx The general solution of the reduced equation is called The COMPLEMENTARY FUNCTION
The REDUCED equation is a
A PARTICULAR INTEGRAL satisfies the equation a
d2y dy b cy f ( x) 2 dx dx
dy by f ( x) is dx the sum of the complementary function and the particular integral yG yP yC
The general solution of a
Solving second order linear differential equations d2y dy b cy f ( x) is a differential equation where a, b and c are real numbers 2 dx dx d2y dy The reduced equation is a 2 b cy 0 dx dx The AUXILIARY equation associated with this equation is a 2 b c 0 The auxiliary equation is a quadratic equation, three cases are possible: a
Case1 : a 2 b c 0 has two distinct solutions 1 and 2 The complementary function is y C1e1x C2 e2 x
C1 , C2
Case 2 : a 2 b c 0 has equal/repeated root 0 The complementary function is y (C1 x C2 )e0 x
C1 , C2
Case 3 : a 2 b c 0 has two conjugate complex solutions 1 p iq and 2 p iq The complementary function is y e px C1Cos (qx) C2 Sin(qx)
C1 , C2
Finding the particular integral: if f ( x) is a polynomial then y P is also a polynomial of the same degree if f ( x) ACos (kx) BSin(kx) then yP aCos (kx) bSin(kx) a and b to be worked out. if f ( x) Ae then yP ae if k kx
kx
or yP axekx if k 1 or 2
where a is to be worked out
or y p ax e if k 0 (the repeated root.) 2 kx
The general solution is yG yP yC
Substitution d2y dy P( x) Q( x) y R( x) is a differential equation 2 dx dx where P,Q and R are functions of x. Note: this equation is written in its standard form. These equations are solved using substitution. The substitution to use will be given in the question.
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Now, consider the numerator of Equation C.4, as follows. Replace the first column of the coefficient matrix [A] with the right-hand side column matrix { f }.
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