Differential Equations Class 12 Mathematics Notes has been updated according to the latest syllabus of 2080. It means the notes of differential equations chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 15
Differential Equations
A differential equation is an equation that involves an unknown function and its derivatives. It represents a relationship between the function and its rate of change, and it plays a critical role in modeling real-world phenomena where quantities change over time or space.
A differential equation involves derivatives of a dependent variable with respect to one or more independent variables. For example, a first-order differential equation is of the form $latex \frac{dy}{dx} = f(x, y)$, where $latex y$ is the dependent variable and $latex x$ is the independent variable.
Differential Equations Class 12 Mathematics Notes PDF
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Order and Degree of Differential Equations
The order of a differential equation is the highest order of derivative present in the equation, while the degree is the power of the highest order derivative, provided the equation is polynomial in derivatives. For example, $latex \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0$ is a second-order differential equation of degree 2.
General and Particular Solutions
The general solution of a differential equation contains arbitrary constants and represents a family of solutions. A particular solution is derived from the general solution by applying specific initial conditions or boundary values.
Formation of a differential equation
A differential equation can be formed from the given equation with the elimination of the constants involved in it. The constants involved in the equation are eliminated by differentiating the given equation and substituting the values of constants in terms of derivative in the given equation. If the equation contains only one arbitrary constant, only one time differentiation is sufficient to eliminate the constant. But if two arbitrary constants are present, we have to differentiate twice for the elimination of two constants.
General solution and Particular solution
A solution of a differential equation is a relation between the dependent and independent variables which is free from derivatives or differentials and which satisfies the given differential equation.SolutIf the number of arbitrary constants is equal to the order of the differential equation, then the solution is known as the general solution.
The solution obtained from the general solution by giving the particular values of the arbitrary constants is known as the particular solution or particular integral.
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