Review of First Order Differential Equations

Differential Equation - An equation that contains the derivative of one or more dependent variables with respect to one or more independent variables is called a differential equation.

• Order of a Differential Equation - It is the degree of the highest derivative appearing in it.
• Degree of a Differential Equation - It is the degree of the highest derivative occurring in it, after the equation has been expressed in a form free from radicals and fractions as far as the derivatives are concerned.

Ordinary Differential Equation (ODE): A differential equation that contains ordinary derivatives of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation.

Partial Differential Equation (PDE): A differential equation that involves partial derivatives with respect to more than one independent variable is called a partial differential equation.

Formation of Differential Equation

Exact Differential Equations

A differential equation of the form $M(x,y)dx + N(x,y)dy = 0$ is said to be exact if its left-hand member is the exact diffrential of some function $u(x,y)$ i.e., $du = Mdx + Ndy = 0$. Its solution, therefore, is the implicit equation $u(x,y) = c$, where $c$ is a constant.

Equation Reducible To Exact Differential Equation

If $Mdx + Ndy = 0$ is a homogeneous equation in x and y , then Integrating Factor $(IF) = \frac{1}{Mx + Ny}$ , provided that $Mx + Ny\: != 0$

Second and Higher Order Linear Differential Equations With Constant Coefficients

Cauchy's & Legendre's Homogeneous Differential Equations

Method of Variation of Parameters

The Method of Variation of Parameters is another technique for finding the particular integral of a linear differential equation when its complementary function is known. This method is general, but it is typically applied to second-order differential equations.

Consider a second-order differential equation in the form:

$\frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = Q(x)$

Where $a$ and $b$ are constants, and $Q(x)$ represents a function of $x$.

The Method of Variation of Parameters is a more versatile approach, but it has some drawbacks. First, it is essential to have the complementary solution to solve the problem. This is in contrast to other methods, such as the method of undetermined coefficients, where having the complementary solution is advisable but not necessary. Second, the method may involve integrals, and it is not guaranteed that these integrals can always be evaluated. Therefore, while it is possible to write down a formula to obtain the particular solution, practical difficulties may arise if the integrals are complex or if the complementary solution cannot be found.