A differential equation relates a function with its derivatives. A differential equation is linear when the dependent variable and its derivatives appear only to the first power, and there are no products between the dependent variable and its derivatives. The general form of a linear differential equation includes coefficients that can be functions of the independent variable. For example, this is a linear differential equation. However, if we have terms like the square of a derivative, the equation becomes non-linear.
First-order linear differential equations have the general form shown here. This is often written in standard form where y prime represents the derivative of y with respect to x. To solve this type of equation, we use an integrating factor, mu of x, which equals e raised to the integral of P of x dx. The general solution is given by this formula, where C is an arbitrary constant. For example, in the equation y prime plus 2x times y equals x, the integrating factor would be e raised to the integral of 2x, which equals e to the power of x squared.
Second-order linear differential equations have the general form shown here. When the right side equals zero, we call it a homogeneous equation. For equations with constant coefficients, we can find solutions using the characteristic equation. Depending on the roots of this equation, we have three cases. If we have two distinct real roots, the solution is a linear combination of exponential functions. If we have a repeated root, the solution includes a term with x multiplied by an exponential. And if we have complex roots, the solution involves exponentials multiplied by sine and cosine functions.
For non-homogeneous linear differential equations, where the right side is not zero, the general solution is the sum of two parts: the homogeneous solution, also called the complementary solution, and a particular solution. The homogeneous solution is found by solving the associated homogeneous equation. To find the particular solution, we can use different methods. The method of undetermined coefficients works when the right side is a polynomial, exponential, sine, cosine, or combinations of these. The variation of parameters method is more general and works for any continuous function on the right side, but it's often more complex to apply.
To summarize what we've learned about linear differential equations: A linear differential equation contains the dependent variable and its derivatives only to the first power, with no products between them. First-order linear equations can be solved using integrating factors. Second-order linear equations with constant coefficients are solved using the characteristic equation, with different solution forms depending on the types of roots. For non-homogeneous equations, the general solution is the sum of the complementary solution and a particular solution. Linear differential equations are fundamental in modeling many physical phenomena in science and engineering, including mechanical vibrations, electrical circuits, and population dynamics.