Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 15.2.5 Green’s functions The Green’s function method of solving linear ODEs bea ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS i.e. the Green’s functionG(x, z)must satisfy the original ODE with the RHS set equa ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS (ii) When considered as a function ofxaloneG(x, z) obeys the specified (homogen ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS y(0) =y(π/2) = 0 is given by y(x)= ∫π/ 2 0 G(x, z)coseczdz =−cosx ∫x 0 sinzcoseczdz ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS
Use Green’s functions to solve d^2 y dx^2 +y=f(x), (15.69) subject to the one- ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS For example, if we consider the second-order case with boundary conditions y(a)=α,y ...

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS where g(x)=a 0 (x)−^14 [a 1 (x)]^2 −^12 a′ 1 (x) h(x)=f(x)exp { 1 2 ∫ a 1 (z)dz ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS 15.3 General ordinary differential equations In this section, we discuss miscellane ...

15.3 GENERAL ORDINARY DIFFERENTIAL EQUATIONS but also write d^2 y dx^2 = dp dx = dy dx dp dy =p dp dy d^3 y dx^3 = d dx ( p dp d ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Solve 2 y d^3 y dx^3 +6 dy dx d^2 y dx^2 =x. (15.80) Directing our attention to th ...

15.3 GENERAL ORDINARY DIFFERENTIAL EQUATIONS 15.3.4 Isobaric or homogeneous equations It is straightforward to generalise the di ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS 15.3.5 Equations homogeneous inxoryalone It will be seen that the intermediate equa ...

15.4 EXERCISES 15.3.6 Equations havingy=Aexas a solution Finally, we note that if any general (linear or non-linear)nth-order OD ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS If the beam is only slightly bent, so that (dy/dx)^2 1, wherey=y(x)isthe downward ...

15.4 EXERCISES 15.9 Find the general solutions of (a) d^3 y dx^3 − 12 dy dx +16y=32x− 8 , (b) d dx ( 1 y dy dx ) +(2acoth 2ax) ( ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS verify that the golden mean is equal to the larger root of the recurrence relation’ ...

15.4 EXERCISES 15.23 Prove that the general solution of (x−2) d^2 y dx^2 +3 dy dx + 4 y x^2 =0 is given by y(x)= 1 (x−2)^2 [ k ( ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS 15.29 The equation of motion for a driven damped harmonic oscillator can be written ...

15.5 HINTS AND ANSWERS 15.36 Find the form of the solutions of the equation dy dx d^3 y dx^3 − 2 ( d^2 y dx^2 ) 2 + ( dy dx ) 2 ...

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS 15.31 Use continuity and the step condition on∂G/∂tatt=t 0 to show that G(t, t 0 )= ...