LECTURE 13, 14
Differential Equations
First Order Equations with Separable Variables (separable equations), First Order Homogeneous Linear Equations, First Order Linear Equations- method of variation of constants, undetermined coefficients, Homogeneous Equations, Second order differential equations
FIRST ORDER DIFFERENTIAL EQUATION
y(t0) = y0 - initial value
Definition
The general solution of a first order differential equation is the family of all functions y(t) that satisfy the equation. Each function in this family is called a particular solution of this equation.
In most cases, the family of functions will depend in some way on a constant C, and the graphs of these functions will form a family of curves that fill up the (t, y) plane, but do not touch each other, as in the Figure 1.
Figure 1
Example:
The general solution of
is:
See the next Figure:
FIRST ORDER DIFFERENTIAL EQUATION WITH SEPARABLE VARIABLES
The graph of the solution:
The above equation has separable variables, because it can be written
The particular solution is x(t) = C t -3 , take y(x) = C(t)t-3
General solution is
The required particular solution is
UNDETERMINED COEFFICIENTS
The method of undetermined coefficients of solving a nonhomogeneous equation
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is based on
Theorem
The general solution of a linear nonhomogeneous differential equation is the sum of a particular solution and the general solution of the homogeneous equation
The general solution of the homogeneous equation is determined as above and the particular solution is guessed.
The guess of the particular solution is relatively easy if :
the function p (x) in the equation is constant (it is then a linear differential equation with constant coefficients), and the function f (x) is:
a polynomial,
a sum α sinω x + β cosω x,
a function aebx, and b ≠ -p,
a sum or a product of the types listed above.
In each of these cases a particular integral of the nonhomogeneous equation should be guessed in the same form as f(x), conserving the degree of a polynomial, as well as numbers and b.
Remaining constants (coefficients of the polynomial, α, β and a) are left undetermined. Their values can be found by back substituting the guess into the nonhomogeneous equation
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It must be noted that this method, unlike the method of variation of coefficients, is not general. However, in the above cases it is computationally simpler.
Example
Using the method of undetermined coefficients we will find a complete integral of the equation
Here p (x) 4 (a constant function), and f (x) = x3 (a polynomial).
1. Find the general solution
The general solution is of the form
2. Find a particular solution using the method of undetermined coefficients
Since
we guess that the particular solution is a polynomial of the 3rd degree, i.e.,
hence
Substituting back into the equation we get
that is
This equality is satisfied for every x if and only if the following system of equations is satisfied
Hence
Thus the function
is the desired particular solution, which implies that
is the general solution of the equation
.
Example
Use the method of undetermined coefficients to the find the particular solution of the equation
with the initial condition y(0) = 2.
It is a nonhomogeneous linear equation of the first order with constant coefficients (p(x) 1), where f(x) = xex .
1. We determine the general solution. It is of the form:
2. We determine the particular solution using the method of undetermined coefficients
We assume it is a product of the polynomial of the first order and the exponential function ex
Hence
Thus
This condition is satisfied if and only if
that is, for any x
Hence
which gives
Thus the particular solution is of the form
whereas the general solution is
Taking into account the initial condition y(0) = 2, we obtain
The desired particular solution is
HOMOGENEOUS EQUATIONS
Let f be a function continuous on an interval (a, b), such that f (u) ≠ u on (a,b).
Definition
A homogeneous differential equation is an equation that can be written in the form
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Equations of this type can be reduced to a separable equation by the substitution
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Indeed, differentiating the equality
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we get
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Substituting in the equation
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Separation of variables yields
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To return to the initial function y ,we substitute u = yx,
Remark
If the condition f (u) ≠ u is not satisfied on (a,b), consider the equation f (u) = u.
Example
We will find the solution to the equation
Substituting
we obtain
and further
For f (u) ≠ u, that is, if
we separate variables and integrate
That is
For f (u) = u we have
Hence the additional solution y ≡ 0.
Quite often, in order to identify an equation to be homogeneous, some transformations are required.
Example
We will solve the equation
Multiplication of the numerator and denominator of the right-hand side of the equation by 1/x2 yields
This above equation is homogeneous.
SECOND ORDER DIFFERENTIAL EQUATION
In order to solve a nonhomogeneous of the second order linear differential equation we have to find a general solution of the corresponding homogeneous equation and then make use of either the method of variation of parameters or the method of undetermined coefficients to find the particular solution.
Example
We will solve the following initial value problem
By the preceding example, the general solution is of the form
We determine the partial solution by use of the method of undetermined coefficients. We guess y1 = Ax2 + Bx +C. Hence y1'' = 2A and
This is satisfied for any x if and only if A = 1, B = 0, C = -2. Then y1 = x2 - 2, and the general solution is of the form
Applying initial conditions we get the constants C1 and C2. We calculate the first derivative
so
Hence C2 = 2, C1 = 1 and the actual solution satisfying the initial conditions reads
The expression "solve an equation" should be understand as a task of determining the general solution of this equation.