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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