How to solve a differential equation?

Overview

A differential equation is an equation consists differential co-efficient.

For example-

  is the DE.

There are two types of differential equations-

Ordinary DE

When a DE consists derivatives with respect to a single independent variable, we call it an ordinary DE.

Partial DE

When a DE consisting partial derivatives with respect to more than one independent variable, we call it a partial DE.

Order and degree of differential equation

The order of the highest differential co-efficient present in the equation is known as the order of a DE.

Degree of a DE is the degree of the highest ordered derivative (when the derivatives are cleared of radicals and fractions).

Note: we form a DE by differentiating the ordinary equation and eliminating the arbitrary constants.

Solution of a differential equation

An equation with dependent variable (y) and independent variable (x) and does not contain derivative, which satisfies the DE, is called the solution of the DE.

Here we will discuss the solution of differential equations with variable separable.

If a DE is

f(y)dy = g(x) dx

Which means, variable are separable, then we can get the solution by using integration.

Steps-

The following steps we follow to find the solution of a differential equation-

1. First we separate the variables such as – f(y)dy = g(x) dx.
2. Now we use integration on both sides-

3. Then finally we add constant C on right hand side.

Solved examples

Example: Solve the following DE.

Solution:

Here we have-

Now separating the variables, we get-

then we integrate both sides-

We get-

Here C is constant.

Example: Solve the DE-

Solution-

Here we have-

Now separating the variables, we get-

Integrate both sides-

Replace the value of v,

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