Cauchy problem and the existence and uniqueness theorem

Bernoulli’s method

Solution by using the change of variable(unknown function)of the form. y = uv, (2)

where u = u (x), v = v (x) are differentiable functions in some domain of x.

Differentiating (2) we get to the conclusion, that the general solution of the inhomogeneous linear differential equation (1) is equal to the sum of general solution of the corresponding homogeneous equation and a particular solution of the inhomogeneous equation (1) Yg.i=Yg.h+Yp.i

Darboux equation is called the equation of the form:

M (x, y) dx + N (x, y) dy + P (x, y)[ xdy − ydx ] = 0 (3), where M and N are homogeneous functions of degree m. P (x, y ) is a homogeneous function of degree l, l ≠ m −1

Darboux equation (3) can be integrated in quadrature with the substitution y = zx, where z is a new unknown function. With this change Darboux equation is reduced to equation with separated variables, if N (x, y) = 0.

Cauchy problem and the existence and uniqueness theorem

Existence theorem. If in the equation у’=ƒ(х,у) the function ƒ is defined and continuous in abounded domain D of the plane (x, y), then for any point (х0, у0)?D exists a solution y (x) of the initial problem

dx/dy = f (x, y), у(х0)=у0, (4) defined on some interval containing х0.

Existence and uniqueness theorem. If the function ƒ is defined and continuous in a bounded domain D of the plane (x, y), and it satisfies in D to Lipschitz condition in the variable y, i.e.

| ƒ(х,у1)- ƒ(х,у2)| ≤ L|у2-у1|, (5) where, L is positive constant, then for any point

(х0,у0)?D exists a unique solution у(х) of the initial problem (4), defined on some interval containing х0.

Extension Theorem. At the conditions of the existence theorem or the theorem of the existence and uniqueness an any solution of the Cauchy problem (3.4) with initial data (х0,у0)?D can be extended to a point arbitrarily close to the boundary of D. In the first case, the continuation, in general, is not necessarily unique; in the second case it is unique.

 

 

4) Bernoulli’s method.

Linear differential equation of the first order is an equation in which the unknown function and its first

derivative are included in the first degree: + p(x)y = q(x) (1)

Bernoulli’s method

Solution by using the change of variable of the form. y = uv, (2)

where u = u (x), v = v (x) are differentiable functions in some domain of x.

Differentiating (2) we get to the conclusion, that the general solution of the inhomogeneous linear differential equation (1) is equal to the sum of general solution of the corresponding homogeneous equation and a particular solution of the inhomogeneous equation (1) Yg.i=Yg.h+Yp.i

This fact is a reflection of the general properties of solutions of linear differential equations. Note that if you can "guess" a particular solution of (1), then the search for general solutions easy.

Darboux equation is called the equation of the form:

M (x, y) dx + N (x, y) dy + P (x, y)[ xdy − ydx ] = 0 (3)

where M and N are homogeneous functions of degree m. P (x, y) is a homogeneous function of degree l, l ≠ m −1

 

 

5) Equations of the first order unsolved by derivatives.

We consider the equation

F (x, y, y’) = 0 (1)

and suppose that it allows the parametric representation

х=ϕ(u, v)

у=ψ(u, v) (2)

у'=α(u, v) so that F(ϕ(u, v),ψ(u, v),α(u, v)) ≡0, for all values of the parameters u and v. We assume, that the functions ϕ(u, v), ψ(u, v), α(u, v) are differentiable. Using the basic relation between the differentials and the derivative along the integral curves of the 1-st order dy=у’dx we find the connection between the parameters u and v. In fact, we have

dx = du+ dv dy= du+ dv у’=α(u, v) Thus, we obtain

du+ dv = α(u, v)[ du+ dv ] (3)

Equation (3) is the equation solved by derivative.

In the equation (3) variables u and v are equal. Taking, for example, u as independent variable, and integrating equation (3), we obtain v = ω(u, c) which is general solution of (3).

 

 

Euler method.

Given a complex- valued solution. It is proved that the real and imaginary parts of the complex - valued solution are solutions. the Euler method which is a method of constructing a fundamental system of solution for the equation

Following Euler we find solution of the equation in the form , where is a constant. We prove that equation has a solution of the form , if satisfies to the equation Where

Equation is called characteristic for . It is considered separately the different cases.

1) The roots of characteristic polynomial are real and different.

2) The roots of characteristic polynomial are different, but among of them there are complex roots

3) The roots of characteristic equation are real, but among of them there multiple.

In the 3^rd case a theorem is proved. To multiple roots multiplicity ‘”k” corresponding to “k” linearly independent solution of the form

4) In the general case can also be multiple complex roots. Note that is the multiplicity of the roots , that multiplicity of the adjoint root as well e. therefore, such pair of roots corresponding to the following linearly independent solutions

 

 

15) Basic properties of linear systems. Vector and matrix form.

 

Linear System Matrix-Vector Form: given an nxn matrix A(t) and nx1 vector valued function f(t):

= a_ik(t)x_k+f_i(t) i=1….n ( pustoi kvadrattyn ornynda ewtene jok)

where a_ik(t),f_i(t) are given functions, x_i(t)is unknown functions called a linear system of differential equations.

We denote

 

Then (1) takes the form =A(t) + (t)

If f(t)=0, the system is homogeneous.if not-it’ll be inhomogeneous

Basic properties of linear system:

Homogeneous System Solution Properties:

-Linear combinations of solutions: if x1(t),x2(t),…xk()t,are solutions to x’=Ax,where x’=dx/dt, then x(t)=C₁x₁(t)+….+C_kx_k(t), is also solution for any constants C₁,…C_k.

-independence:suppose y1(t),y2(t),…y_k(t), are solutions to y’=Ay,for tє I=(a,b)

a)if y₁(t₀)..y_k(t_o) are dependent for some t₀ є I,then there is exist C1,C2…Ck,not all 0,so that C1y1(t)+C2y2(t)+….+C_k y_k (t)=0 t? I

(the y_i(t)'s are dependent any t є I);

b) if y₁(t₀)..y_k(t_o) are dependent for some t₀ є I,then y_i(t)'s are dependent any t є I).

-Solution Structure: if y1(t)….yn(t),are linearly independent solutions to the n-dimensioanl system y’=Ay, then any solution has the form y(t)= C1y1(t)+C2y2(t)+….+C_ny_n (t) for some constants C₁,C₂….C_n.

The n y_i's form a fundamental set of solutions.

- Solution Strategy:

a)find n independent y_i's to form general solution;

b) if initial value y(t₀) is given,

solve Y(to)C=y(t0) for C=(C1,C2….Cn)^T, using nxn matrix we have Wronskian

 

 

y_11….. y_1n this expression called

y₂₁….. y_2n the wronskian

….

Y_n1….. y_nn

 

 

Orbital stability

The orbital stability differs from the Lyapunov stabilities in that it concerns with the stability of a system output (or state) trajectory under small external perturbations. n

Let f(x) be a p-periodic solution, p>0, of the autonomous system x(t)=f(x), x(t0)=x0?۠۠۠R and let Г represent the closed orbit of f(x) in the state space, namely, Г={y|y=f(x0),0≤t<p}

If, for any e>0, there exists a constant b=b(e)>0 such that d(x0, Г):=inf||x0-y||<b the solution of the system, f(x) satisfies d(f(x), Г) < e, for all t≥t0 then this p-periodic solution trajectory, f(x) is said to be orbitally stable.

 

The main Lyapunov theorems.

Basic theorem of Lyapunov

Let V(x,t) be a non-negative function with derivative V’ along the trajectories of the system.

1.If V(x,t) is locally positive definite and V’(x,t) ≤0 locally in x and for all t, then the origin of the system is locally stable(in the sense of Lyapunov)

2.If V(x,t) is locally positive definite and decrescent,and V’(x,t) ≤0 locally in x and for all t, then the origin of the system is uniformly locally stable(in the sense of Lyapunov)

3. If V(x,t) is locally positive definite and decrescent, and -V’(x,t) ≤0 locally positive definite, then the origin of the system is uniformly locally asymptotically stable

4.If V(x,t) is locally positive definite and decrescent, and -V’(x,t) ≤0 locally positive definite, then the origin of the system is uniformly locally asymptotically stable

Theorem. Exponential stability theorem

x*=0 is an exponentially stable equilibrium point of x’=f(x,t) if and only if there exists an e>0 and a function V(x,t) which satisfies

 

29.Chetaev theorem nn n

Theorem. [adapted from Chetaev 1934].Suppose E? R is open, 0? E, f;E→R and V: E→R are continuously differentiable, f(0)=0, V(0)=0. Suppose U an open neighborhood of 0 with compact closure F=U?E such that the restriction of V’=(DV)f to F∩V pow-1(0,∞) is strictly positive. If for every open neighborhood W?R pow n of 0 the set W∩W pow -1 (0,∞) is nonempty, then the origin is an unstable equilibrium of x’=f(x).

Proof:

1.Assume above hypotheses

2.Let W be an open neighborhood of 0?R pow n

3There exists z?W∩V pow -1(0,∞).Define a=V(z)>0

4.The set K=F∩ V pow -1[a,∞) is compact, and it is nonempty since z?K

5Using the contiuity of V and V’ there exist m=minV’(x) and M=maxV(x)

6 Since K=F∩ V pow -1[a,∞)? F∩ V pow -1[a,∞) it follows that m>0

7 Let I? [0,∞) be the maximum interval of existence for empty set:I→E such that empty(0)=z

8 Define the set T={t? I: for all s?[0,t], empty set(s)=K }

9.For all t? T d/dt(V*empty set)(t)=(V’*empty set)(t)>m

10Hence for all t? T V(empty set(t))=V(empty set(0))+∫V(empty(s))ds≥a+mt

11 Since V is bounded above by M,T is also bounded above.Let t0 the least bounded of T.

12 Since K?E,I [0,t0] and empty(t0) is well-defined. Moreover t0 lies in the interior of I.

13 Since for all t?[0,t0),empty(t0)?K and for every b>0 there exist t?I,t0<t<t0+b such that it follows that empty(t0) lies on the boundary of the set K.

14 The boundary of K is contained in the union ӘK?V pow -1(a)U (F\U)

15 Since V(empty(t0))>a+m*t0>a it follows that empty(t0)? not V(a)pow -1

16 Therefore empty(t0)?(F\U) and in particular empty(t0)? not U

17 Since W was arbitrary this shows that the origin is an unstable equilibrium point of f.

 

Local and global theorems.

Theorem. For any point T there exists a fundamental matrix solution defined in some small neighborhood U of .

Proof. The linear vector-function (t, x) is holomorphic everywhere on T × . By the local existence theorem, for any initial condition (, ) T × there exists a holomorphic vector solution defined on some neighborhood of , meeting the condition . Choose n solutions satisfying n linear independent initial conditions at , arranged as columns of a square matrix and considered on their common domain.

By construction, , hence the holomorphic matrix is holomorphically invertible in some neighborhood of the point .

Theorem. (global existence theorem). A linear system on a Riemann surface T admits a fundamental solution in any simply connected subdomain

Proof. Choose a base point and let be a local fundamental matrix solution at this point. We extend it to an arbitrary point .

Since is connected, there exists a compact piecewise smooth curve (path) γ connecting with . γ can be covered by carrying the respective local fundamental matrix solutions , such that are connected, and if and only if |i − j| 1.

Assume that satisfy , , . Then ,

Agree on the intersections:

,

the solution can be explicitly constructed:

.

This completes the proof of existence of analytic continuation of solutions along paths.

 

Liouville formula.

Definition. Let are real-valued functions on , which are times differentiable on . Then their Wronskian is defined by

 

Theorem (The Liouville formula) Let be a sequence of solutions of is continuous. Then the Wronskian of this sequence satisfies the identity

for all .

Recall that the trace trace A of the matrix A is the sum of all the diagonal entries of the matrix.

Proof. Let the entries of the matrix ( then

We use the following formula for differentiation of the determinant, which follows from the full expansion of the determinant and the product rule:

Indeed, if are real-valued differentiable functions then the product rule implies by induction

Hence, when differentiating the full expansion of the determinant, each term of the determinant gives rise to n terms where one of the multiples is replaced by its derivative. Combining properly all such terms, we obtain the derivative of the determinant is the sum of n determinants where one of the rows is replaced by its derivative, that is, (1).

The fact that each vector satisfies the equation can be written in the coordinate form as follows

 

 

Liouville formula.

Theorem (The Liouville formula) Let be a sequence of solutions of is continuous. Then the Wronskian of this sequence satisfies the identity

for all .

Recall that the trace trace A of the matrix A is the sum of all the diagonal entries of the matrix.

Proof. Let the entries of the matrix ( then

We use the following formula for differentiation of the determinant, which follows from the full expansion of the determinant and the product rule:

Indeed, if are real-valued differentiable functions then the product rule implies by induction

 

 

Hence, when differentiating the full expansion of the determinant, each term of the determinant gives rise to n terms where one of the multiples is replaced by its derivative. Combining properly all such terms, we obtain the derivative of the determinant is the sum of n determinants where one of the rows is replaced by its derivative, that is, (1).

The fact that each vector satisfies the equation can be written in the coordinate form as follows

 

Bernoulli’s method

Solution by using the change of variable(unknown function)of the form. y = uv, (2)

where u = u (x), v = v (x) are differentiable functions in some domain of x.

Differentiating (2) we get to the conclusion, that the general solution of the inhomogeneous linear differential equation (1) is equal to the sum of general solution of the corresponding homogeneous equation and a particular solution of the inhomogeneous equation (1) Yg.i=Yg.h+Yp.i

Darboux equation is called the equation of the form:

M (x, y) dx + N (x, y) dy + P (x, y)[ xdy − ydx ] = 0 (3), where M and N are homogeneous functions of degree m. P (x, y ) is a homogeneous function of degree l, l ≠ m −1

Darboux equation (3) can be integrated in quadrature with the substitution y = zx, where z is a new unknown function. With this change Darboux equation is reduced to equation with separated variables, if N (x, y) = 0.

Cauchy problem and the existence and uniqueness theorem

Existence theorem. If in the equation у’=ƒ(х,у) the function ƒ is defined and continuous in abounded domain D of the plane (x, y), then for any point (х0, у0)?D exists a solution y (x) of the initial problem

dx/dy = f (x, y), у(х0)=у0, (4) defined on some interval containing х0.

Existence and uniqueness theorem. If the function ƒ is defined and continuous in a bounded domain D of the plane (x, y), and it satisfies in D to Lipschitz condition in the variable y, i.e.

| ƒ(х,у1)- ƒ(х,у2)| ≤ L|у2-у1|, (5) where, L is positive constant, then for any point

(х0,у0)?D exists a unique solution у(х) of the initial problem (4), defined on some interval containing х0.

Extension Theorem. At the conditions of the existence theorem or the theorem of the existence and uniqueness an any solution of the Cauchy problem (3.4) with initial data (х0,у0)?D can be extended to a point arbitrarily close to the boundary of D. In the first case, the continuation, in general, is not necessarily unique; in the second case it is unique.

 

 

4) Bernoulli’s method.

Linear differential equation of the first order is an equation in which the unknown function and its first

derivative are included in the first degree: + p(x)y = q(x) (1)

Bernoulli’s method

Solution by using the change of variable of the form. y = uv, (2)

where u = u (x), v = v (x) are differentiable functions in some domain of x.

Differentiating (2) we get to the conclusion, that the general solution of the inhomogeneous linear differential equation (1) is equal to the sum of general solution of the corresponding homogeneous equation and a particular solution of the inhomogeneous equation (1) Yg.i=Yg.h+Yp.i

This fact is a reflection of the general properties of solutions of linear differential equations. Note that if you can "guess" a particular solution of (1), then the search for general solutions easy.

Darboux equation is called the equation of the form:

M (x, y) dx + N (x, y) dy + P (x, y)[ xdy − ydx ] = 0 (3)

where M and N are homogeneous functions of degree m. P (x, y) is a homogeneous function of degree l, l ≠ m −1

 

 

5) Equations of the first order unsolved by derivatives.

We consider the equation

F (x, y, y’) = 0 (1)

and suppose that it allows the parametric representation

х=ϕ(u, v)

у=ψ(u, v) (2)

у'=α(u, v) so that F(ϕ(u, v),ψ(u, v),α(u, v)) ≡0, for all values of the parameters u and v. We assume, that the functions ϕ(u, v), ψ(u, v), α(u, v) are differentiable. Using the basic relation between the differentials and the derivative along the integral curves of the 1-st order dy=у’dx we find the connection between the parameters u and v. In fact, we have

dx = du+ dv dy= du+ dv у’=α(u, v) Thus, we obtain

du+ dv = α(u, v)[ du+ dv ] (3)

Equation (3) is the equation solved by derivative.

In the equation (3) variables u and v are equal. Taking, for example, u as independent variable, and integrating equation (3), we obtain v = ω(u, c) which is general solution of (3).