Unformatted text preview: MEC702 Wk 6 Lect 2 Higher Order Linear ODEs
Homogneneous Linear ODEs • The ODE is of nth order if it includes the nth derivative
y (n) = d4 y
dx4 of the unknown functiony(x) in the Ordinary Dierential Equation. Thus
the ODE is of the form
F (x, y, y 0 , · · · , y (n) ) = 0 where the lower derivatives and y itself may or may not occur. Such an
ODE is called linear if it can be written
y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = r(x) The coecients p0 , · · · , pn−1 and the function r on the right are the given
functions of x.
• y and all its derivatives y (n) have the coecient 1.
• All nthorder ODE that can not be written in the form above is called . nonlinear • If r(x) = 0, then
y (n) + pn−1 (x)y (n−1) + · · · + p1 )xy 0 + p0 (x)y = 0 is homogeneous. If r(x) ≡ 0, then the ODE is called nonhomogeneous.
• A solution of an nthorder (linear or linear) ODE on some open interval
I is a function
y = h(x) that is dened andn times dierentiable on I 1 Homogeneous Linear ODE: Superposition Principle, General Solution .The basic superposition principle (given for 2nd order homogeneous linear
ODE) extends to nth order homogenous linear ODEs as follows:
Theorem. (Fundamental Theorem for the Homogeneous Linear ODE) For a
homogeneous linear ODE of nth order, the sums and constant multiples of soI are again solutions on I . (This does not hold lutions on some open interval for any nonhomogeneous or nonlinear ODE!!) Theorem. (General Solution, Basis, Particular Solution) A general solution of y (n) + pn−1 (x)y (n−1) + · · · + p1 )xy 0 + p0 (x)y = 0
on an open interval I is a solution of the ODE on I of the form y(x) = c1 y1 (x) + · · · + cn yn (x)
where I; y1 , · · · , yn is a basis (or fundamental system) of solutions of the ODE on that is, these solutions are linearly independent on I. • A particular solution of the above ODE on I is obtained if we assign
specic values to the n constraints c1 , . . . , cn in the solution. IVP and n Initial Conditions An initial value problem for the nth order linear homogeneous ODE consists of
n initial conditions
y(x0 ) = K0 , y 0 (x0 ) = K1 , . . . , y (n−1) (x0 ) = Kn−1 with given x0 , K0 , K1 , . . . , Kn−1 . 2 Extension of Methods from 2nd order ODE to nth order ODE The following are the methods we learnt in solving ODEs of 2nd order:
• λmethod  solving homogeneous linear of 2nd order ODE
• EulerCauchy Equations of 2nd order
• Wronkian  testing linear independence of basis of solutions for 2nd order ODE • Method of Undetermined Coecients  solving linear nonhomogeneous
ODE of 2nd order
• Method of Variation of Parameters  solving linear nonhomogeneous ODE
of 2nd order
Note that each of these methods can be extended to solving ODE of nth order!!!! Example. 1. Solve the fourthorder ODE
y (4) − 5y 00 + 4y = 0. 2. Solve y 000 − 2y 00 − y 0 + 2y = 0. 3. Solve the IVP
y 000 − y 00 + 100y 0 − 100y = 0, 4. Solve the ODE y(0) = 4, y 0 (0) = 11, y 00 (0) = −299. y v − 3y iv + 3y 000 − y 00 = 0. 5. Solve the IVP
x3 y 000 − 3x2 y 00 + 6xy 0 − 6y = 0, y(1) = 2, y 0 (1) = 1, 6. Check whether the functions
y1 = e−2x , y2 = e−x , are linearly independent or dependent. 3 y3 = ex y 00 (1) = −4. ...
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 Winter '16
 Alveen Chand
 Derivative, Ode

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