91Ó°ÊÓ

A linear homogeneous differential equation is a type of first-order differential equation. Such equations can be described in the form: \[ \frac{dy}{dx} + P(x)y = 0 \] The differential equation given in the problem, \( \frac{dy}{dx} + 2xy = 0 \), is an example of this kind of equation where \( P(x) = 2x \). The term "linear" implies that the equation involves only the first power of \( y \), and "homogeneous" indicates that there is no constant term (or function) on the other side of the equation, leaving it equal to zero.
Characteristics of linear homogeneous differential equations include:

• The derivative of \( y \) appears only to the first degree.
• The coefficients of \( y \) and \( dy/dx \) are functions of \( x \).
• No terms are free of \( y \) or its derivatives.

This type of equation allows for the method of integrating factors to be effectively applied in finding solutions.

To solve a linear homogeneous differential equation, we often use an integrating factor. This is a function which, when multiplied with the differential equation, transforms it into a form that is easily integrable. For the general equation: \[ \frac{dy}{dx} + P(x)y = 0 \] the integrating factor \( \mu(x) \) is defined as: \[ \mu(x) = e^{\int P(x) \, dx} \] In our exercise, since \( P(x) = 2x \), the integrating factor becomes: \[ \mu(x) = e^{\int 2x \, dx} = e^{x^2} \]
By multiplying the original differential equation by this integrating factor, we get: \[ e^{x^2} \left( \frac{d y}{d x} + 2xy \right) = 0 \] This manipulation transforms the original equation such that it becomes straightforward to integrate both sides.

After applying the integrating factor and transforming the differential equation, integrating both sides gives a solution related to an expression in terms of \( y \) and \( x \). For example, solving the transformed equation: \[ \frac{d}{dx}(y e^{x^2}) = 0 \] integrating gives: \[ y e^{x^2} = C \] where \( C \) is an arbitrary constant. This represents the general solution in implicit form.
If necessary, we can convert this implicit solution into an explicit form by solving for \( y \): \[ y = \frac{C}{e^{x^2}} = Ce^{-x^2} \] This is a clear, beautiful representation of the solution that can be used to find specific solutions if initial conditions are given.
General solutions in first-order linear homogeneous differential equations often contain this arbitrary constant \( C \), allowing us to adjust to various initial conditions and find specific solutions as needed.