Answer to Question #243024 in Differential Equations for Agape

Question

Consider the linear first order equation
x

Answer 1

Definition of Linear Equation of First Order

A differential equation of type $y′+a(x)y=f(x),$ 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: $y′+a(x)y=f(x),$ the integrating factor is defined by the formula

$u(x)=e^{(∫a(x)dx)}.$ Multiplying the left side of the equation by the integrating factor u(x)

 converts the left side into the derivative of the product y(x)u(x).

The general solution of the differential equation is expressed as follows:

$y=∫u(x)f(x)dx+Cu(x),$ 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:

y′+a(x)y=0.

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.

Initial Value Problem

If besides the differential equation, there is also an initial condition in the form of $y(x_0)=y_0$ ,

 such a problem is called the initial value problem (IVP) or Cauchy problem.

A particular solution for an IVP does not contain the constant C,

 which is defined by the substitution of the general solution into the initial condition