Differential Equations

Types of Differential Equations

•    Ordinary Differential Equations

•   Partial Differential Equations

•    Linear Differential Equations

•    Nonlinear differential equations

•    Homogeneous Differential Equations

•    Nonhomogeneous Differential Equations

Equation of first order and first degree

In this section we will consider,

The equation of the form M + N (dy/dx) = 0; where M and N are both function of x and y.

This equation is often written as;

Mdx + Ndy = 0

Methods of solving equation of first order and first degree

•        Separation of Variables

•         Integrating Factors

•         Exact Equations

•         Homogeneous Equations

•         Linear Equations

•         Exact Differential Equations

Separation of Variables

• In this method, the equation is rearranged so that all terms involving the dependent variable y are on one side and all terms involving the independent variable x are on the other side. Then, you integrate both sides with respect to their respective variables.

In the given equation M(x,y)dx + N(x,y)dy = 0 can written in the form f(x)dx = g(y)dy.

Integrating Factors

• This method is useful when the equation is not directly separable. An integrating factor is multiplied to both sides of the equation to make it exact, thus allowing for easier integration.

Exact Equations

• Some first-order differential equations can be written in a form where they are exact differentials. In such cases, the solution involves finding a function whose total differential matches that of the given equation.

Homogeneous Equations

• If the equation can be written in a homogeneous form (we can test whether a function is homogeneous by taking y = vx. ) , where all terms have the same degree, it can be solved using a substitution to reduce it to a separable form.

Bernoulli Equations

• These equations can be transformed into linear equations by using a suitable substitution, typically z = y^(1-n), where n is a constant.

Linear Equations

• If the first-order equation can be expressed in the form dy/dx + P(x)y = Q(x), it's a linear first-order differential equation. These equations can be solved using integrating factors or by using specific formulas depending on the coefficients P(x) and Q(x).

 Exact Differential Equations

• Some differential equations can be written in a form where they are exact differentials. In such cases, the solution involves finding a function whose total differential matches that of the given equation.