
- •Differential equation of first order. Separable Equations
- •2. Homogeneous Equations. Linear de of first order.
- •3. Bernoulli Equation. Singular Solution
- •4. Ricatti Equation.
- •5. Exact Equations
- •Using an Integrating Factor
- •Existence and uniqueness of solution of Cauchy problem for de of first order and system of equations.
- •Implicit Equation. Introducing the parameter.
- •Singular Solutions.
- •Lagrange and Clairaut Equations.
- •Reduction of Order.
- •Linear homogeneous and non-homogeneous de of n-th order
- •The theorem of existence and uniqueness of the solution for Linear de
- •Linear independent system of functions.
- •Wronskian.
- •Liouville's formula
- •Structure of the General Solution
- •Reduction of order
- •Characteristic equation. The roots of the Characteristic Equation are Real and Multiple, Complex and Distinct, Complex and Multiple.
- •Method of Variation of Parameters
- •Method of Undetermined Coefficients
- •Linear Homogeneous Systems of de with variable coefficients
- •26) Fundamental System of Solutions and Fundamental Matrix
- •27. Wronskian and Liouville’s Formula.
- •28. Method of Variation of Constants (Lagrange Method)
- •29. Linear nonhomogeneous systems of differential equations with constant coefficients.
Differential equation of first order. Separable Equations
A first order differential equation y' = f(x,y) is called a separable equation if the function f(x,y) can be factored into the product of two functions of x and y:
where p(x) and h(y) are continuous functions.
Considering
the derivative y' as
the ratio of two differentials
we
move dx to
the right side and divide the equation by h(y):
Of
course, we need to make sure that h(y)
≠ 0. If there's a number x0 such
that h(x0)
= 0, then this number will also be a solution of the differential
equation. Division by h(y) causes
loss of this solution.
Denoting
,
we write the equation in the form
We have separated the variables so now we can integrate this equation:
where C is an integration constant. Calculating the integrals, we get the expression
representing the general solution of the separable differential equation.
Example:
1
We can represent the constant C as lnC1, where C1 > 0. Then
Thus, the given differential equation has the following solutions:
This answer can be simplified. Indeed, if using an arbitrary constant C, which takes values from −∞ to +∞, the solution can be written in the form:
When C = 0, it becomes y = 0.
2. Homogeneous Equations. Linear de of first order.
A first order differential equation
is called homogeneous equation, if the right side satisfies the condition
for all t. In other words, the right side is a homogeneous function (with respect to the variables x and y) of the zero order:
Definition of Linear Equation of First Order
A differential equation of type
where a(x) and f(x) are continuous functions of x, is called a linear nonhomogeneous differential equation of first order.
We consider two methods of solving linear differential equations of first order:
Using an integrating factor;
Method of variation of a constant.
Using an Integrating Factor
If a linear differential equation is written in the standard form:
the integrating factor is defined by the formula
Multiplying the left side of the equation by the integrating factor u(x) converts the left side into the derivative of the producty(x)u(x). The general solution of the differential equation is expressed as follows:
where C is an arbitrary constant.
Method of Variation of a Constant
This method is similar to the previous approach. First it's necessary to find the general solution of the homogeneous equation:
The general solution of the homogeneous equation contains a constant of integration C. We replace the constant C with a certain (still unknown) function C(x). By substituting this solution into the nonhomogeneous differential equation, we can determine the function C(x). The described algorithm is called the method of variation of a constant. Of course, both methods lead to the same solution.