Mechanical Engineers Handbook Episode 14 pps

c-1.4 Linear Differential Equations 1.4.1 HOMOGENEOUS LINEAR EQUATIONS: DEFINITIONS AND GENERAL PROPERTIES DEFINITION 1.1 An n-order differential equation is called linear if it is of th

EXAMPLE 1.33 Find the ®rst three successive approximations y1, y2, y3for the solution of the

equation dy=dx ˆ 1 ‡ xy2, y…0† ˆ 0, in a rectangle D: ÿ1

We can write that f …x; y† ˆ 1 ‡ xy2, maxDj f …x; y†j ˆ 1 ‡1

jy ÿ y3j ML4!3h4ˆ98 12

 3 49

 414!

8

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SolutionThe functions f …x; y; z† ˆ z and g…x; y; z† ˆ ÿy2 are continuous in D.maxDj f …x; y; z†j ˆ maxDjzj ˆ 1; maxDjg…x; y; z†j ˆ maxDj ÿy2j ˆ 1, that

Lg ˆ 2:

h ˆ min a;b1

M;

b2M

y1…x† ˆ

…x

0 z0…t†dt ˆ

…x0

dt ˆ1

2ÿ

x312

y3…x† ˆ

…x

0 z2…t†dt ˆ

…x0

1

2ÿ

t312

y0ˆ fi…x; y†; …i ˆ 1; 2; †, and if each of the equations y0ˆ fi…x; y† in theneighborhood of point (x0; y0) satis®es the conditions of the theorem ofexistence and uniqueness, then for each one of these equations there will be

a unique solution satisfying the condition y…x0† ˆ y0

THEOREM 1.6 There is a unique solution y ˆ y…x†, x0ÿ h  x  x0‡ h (with h0

suf®-ciently small) of Eq (1.126) that satis®es the condition y…x0† ˆ y0, for which

y0…x0† ˆ y0

0, where y0

0is one of the real roots of the equation F …x0; y0; y0† ˆ 0

if in a closed neighborhood of the point (x0; y0; y0), the function F …x; y; y0†satis®es the following conditions:

1.F…x; y; y0† is continuous with respect to all arguments

2.The derivative @F=@y0exists and is nonzero in (x0; y0; y0†

3.There exists the derivative @F=@y bounded in absolute value

@F=@y

Remark 1.19The uniqueness property for the solution of equation F …x; y; y0† ˆ 0, whichsatis®es condition y…x0† ˆ y0, is usually understood in the sense that notmore than one integral curve of equation F …x; y; y0† ˆ 0 passes through agiven point (x0; y0) in a given direction.

EXAMPLE 1.35 Let us consider the problem xy02ÿ 2yy0ÿ x ˆ 0; y…1† ˆ 0

(a) Study the application of Theorem 1.6

(b) Solve the problem m

Solution(a) The function F …x; y; y0† ˆ xy02ÿ 2yy0ÿ x is continuous with respect toall arguments and x0ˆ 1, y0ˆ 0 F …x0; y0; y0† ˆ 0 ) F …1; 0; y0† ˆ 0, that is,

y02ÿ 1 ˆ 0 ) y0

01ˆ 1 and y0

02ˆ ÿ1 @F =@y ˆ 2xy0ÿ 2y; @F =@y0…1; 0; 1† ˆ

2 6ˆ 0; @F =@y0…1; 0; ÿ1† ˆ ÿ2 6ˆ 0 Hence, the derivative exists and is zero @F =@y ˆ ÿ2y0; @F =@y ˆ jÿ2y0j  M1 The considered problem has aunique solution

y ÿpy2‡ x2xwith the solutions y ˆ1

(b) Solve the problem m

Solution(a) The function F …x; y; y0† ˆ 2y02‡ xy0ÿ y is continuous and @F =@y ˆÿ1 ) @F =@y ˆ 1 is bounded @F=@y0ˆ 4y0‡ x, x0ˆ 2, y0ˆ ÿ1

(b) The equation 2y02‡ xy0ÿ y ˆ 0 has the general solution

y ˆ cx ‡ 2c2 and the singular solution y ˆ ÿ…x2=8† Condition y…2† ˆ ÿ1

2

solu-1.3.3 SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS

The set of points (x; y) at which the uniqueness of solutions for equation

F …x; y; p† ˆ 0 and @F =@p ˆ 0 If a branch y ˆ j…x† of the curve F…x; y† ˆ 0belongs to the singular set and at the same time is an integral curve, it iscalled a singular integral curve, and the function y ˆ j…x† is called asingular solution Thus, in order to ®nd the singular solution of Eq (1.127)

it is necessary to ®nd the PDC de®ned by the equations F …x; y; p† ˆ 0,

@F =@p ˆ 0, to ®nd out [by direct substitutions into Eq (1.127)] whether thereare integral curves among the branches of the PDC and, if there are suchcurves, to verify whether uniqueness is violated in the points of these curves

or not If the uniqueness is violated, then such a branch of the PDC is asingular integral curve The envelope of the family of curves

envelope forms a part of the c-discriminant curves (CDCs) determined by thesystem

@y

exx x1

ˆ ex…1 ÿ x† 6ˆ 0; x 2 ‰2; 1†:

The solution y1, y2is linearly independent on [2; 1) Using Eq (1.147) gives

y0ˆ y0 1

…udx ‡ y1u ˆ ex…

…2x ÿ 3†u00ÿ 2u0ÿ …2x ÿ 5†u ˆ 0:

u1ˆ exx 0ˆ …xeÿx†0ˆ …1 ÿ x†eÿx

u ˆ u1

…vdx ˆ …1 ÿ x†eÿx…

vdx

u0ˆ u0 1

…vdx ‡ u1v ˆ …x ÿ 2†eÿx…

v…x† ˆe2x…2x ÿ 3†

…x ÿ 1†2 :Substituting this expression in u ˆ u1„vdx gives

…ÿex†dx ˆ ÿe2x If y ˆ ÿe2x is a solution of ahomogeneous equation, then y3…x† ˆ e2x is also a solution of that equation

The system of functions fex; x; e2xg is a fundamental system …W ‰ y1y2; y3Š ˆ

ÿ…2x ÿ 3† 6ˆ 0; 8x 2 ‰2; 1††, and the general solution is

y…x† ˆ c1ex ‡ c2x ‡ c3e2x:

EXAMPLE 1.43 Find the homogeneous linear differential equation, knowing that the

funda-mental system of solutions is x, ex, e2x m

SolutionUsing Remark 1.26, this equation is

ˆ 0;

...

ayi…x†yj…x†dx; i; j ˆ 1; 2; ; n: …1 :141 †The determinant

…1 :142 †

is called the Grammian of the system of functions fyk…x†g... a1…x†y…nÿ1†‡


‡ an…x†y ˆ …1 :148 †

y…n†‡ b1…x†y…nÿ1†‡


‡ bn…x†y ˆ 0; …1 :149 †where the functions ai…x† and...

W …x† ˆ W …x0†eÿ

„x x0 a …x†dx

Equation (1 .143 ) is called Liouville''s formula m

THEOREM 1.9 If the Wronskian W …x† ˆ