Notes On Solutions of a Differential Equation - CBSE Class 12 Maths

Consider the differential equation dy/dx = ex - 4

We claim that function y = ex - 4x + 3 is the solution of the differential equation.

The solution of a differential equation is the function that satisfies it.


There exist two types of solutions for a differential equation:

1. General solution

2. Particular solution

dy/dx = ex - 4

y = ex - 4x + 3

Suppose function y is of the forms as shown.

y = ex - 4x + 5 ⇒ dy/dx = ex - 4

y = ex - 4x - 5  ⇒ dy/dx = ex - 4

y =ex - 4x + 122 ⇒ dy/dx = ex - 4

Each of the three forms of the function y entitles to be the solution of the differential equation, since their derivatives are the same.

y = ex - 4x + C

We can represent the constant in the equations with a letter. We chose C here.

So, this function represents all the solutions of the differential equation. Such a function is called the general solution.

And, the function obtained by replacing the value of 'C' by a number is called the particular solution.

Summary

Consider the differential equation dy/dx = ex - 4

We claim that function y = ex - 4x + 3 is the solution of the differential equation.

The solution of a differential equation is the function that satisfies it.


There exist two types of solutions for a differential equation:

1. General solution

2. Particular solution

dy/dx = ex - 4

y = ex - 4x + 3

Suppose function y is of the forms as shown.

y = ex - 4x + 5 ⇒ dy/dx = ex - 4

y = ex - 4x - 5  ⇒ dy/dx = ex - 4

y =ex - 4x + 122 ⇒ dy/dx = ex - 4

Each of the three forms of the function y entitles to be the solution of the differential equation, since their derivatives are the same.

y = ex - 4x + C

We can represent the constant in the equations with a letter. We chose C here.

So, this function represents all the solutions of the differential equation. Such a function is called the general solution.

And, the function obtained by replacing the value of 'C' by a number is called the particular solution.

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