Key Points

A differential equation is an equation that involves an unknown function and one or more of its derivatives. It relates a function with its derivatives.

The order of a differential equation is the order of the highest derivative appearing in the equation. For example, the order of $\frac{d^3y}{dx^3} + y = 0$ is 3.

The degree is the highest power (positive integer) of the highest order derivative, provided the equation is a polynomial in its derivatives. For $(\frac{d^2y}{dx^2})^3 + \frac{dy}{dx} = 0$, the degree is 3.

The degree of a differential equation is not defined if it cannot be expressed as a polynomial in its derivatives. For example, the degree of $\frac{d^2y}{dx^2} + \sin(\frac{dy}{dx}) = 0$ is not defined.

A general solution contains arbitrary constants equal in number to the order of the equation. A particular solution is obtained by giving specific values to these constants, usually from initial conditions.

For a differential equation of the form $\frac{dy}{dx} = f(x)g(y)$, we can separate variables to get $\frac{1}{g(y)}dy = f(x)dx$. The solution is found by integrating both sides: $\int \frac{1}{g(y)}dy = \int f(x)dx + C$.

A differential equation is homogeneous if it can be written in the form $\frac{dy}{dx} = F(\frac{y}{x})$. To solve, substitute $y = vx$, which implies $\frac{dy}{dx} = v + x\frac{dv}{dx}$.

After substituting $y=vx$, the equation becomes $v + x\frac{dv}{dx} = F(v)$. This can be rearranged into a variable separable form: $\frac{dv}{F(v) - v} = \frac{dx}{x}$, which can then be integrated.

A first-order linear differential equation has the form $\frac{dy}{dx} + Py = Q$, where P and Q are constants or functions of $x$ only.

To solve a linear differential equation, first find the Integrating Factor (I.F.), which is given by the formula $I.F. = e^{\int P dx}$.

The general solution of the linear differential equation $\frac{dy}{dx} + Py = Q$ is given by $y \times (I.F.) = \int (Q \times I.F.) dx + C$.

Another form of a linear differential equation is $\frac{dx}{dy} + P_1x = Q_1$, where $P_1$ and $Q_1$ are constants or functions of $y$ only.

For the form $\frac{dx}{dy} + P_1x = Q_1$, the Integrating Factor is $I.F. = e^{\int P_1 dy}$. The solution is given by $x \times (I.F.) = \int (Q_1 \times I.F.) dy + C$.