Differential Equations Class 12 Mathematics Solutions 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
In Class 12 Mathematics, the study of differential equations of the first order and first degree is a fundamental topic. A differential equation of the first order involves the first derivative of the dependent variable with respect to the independent variable, and being of the first degree means that the highest power of the derivative in the equation is one.
The differential equations of the first order and first degree are foundational in understanding more complex differential equations. They are simple yet powerful tools for modeling and solving real-world problems where a rate of change is involved.
Differential Equations Class 12 Mathematics Solutions PDF
This PDF will provide the solutions of every question from the 2nd exercise of class 12 differential equation chapter. If you want the notes of other exercises then you can choose the exercise from the button given above.
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General Form
A first-order and first-degree differential equation is typically written in the form: $latex \frac{dy}{dx} + P(x)y = Q(x)$ Here, $latex y$ is the dependent variable, $latex x$ is the independent variable, $latex \frac{dy}{dx}$ is the first derivative of $latex y$ with respect to $latex x$, and $latex P(x)$ and $latex Q(x)$ are functions of $latex x$.
Solution Methods
- Variable Separation: This method is applicable when the equation can be written such that all terms involving $latex y$ are on one side and all terms involving $latex x$ are on the other. The general form for separable variables is: $latex g(y) , dy = f(x) , dx$ Integrating both sides yields the solution.
- Homogeneous Differential Equations: These are equations where the function $latex P(x, y)$ is homogeneous, meaning it can be expressed as a function of $latex \frac{y}{x}$. Substituting $latex y = vx$ (where $latex v$ is a function of $latex x$) converts the equation into a separable form.
- Linear Differential Equations: These take the form: $latex \frac{dy}{dx} + P(x)y = Q(x)$ . The solution is found using an integrating factor, which is given by: $latex \text{Integrating Factor (I.F.)} = e^{\int P(x)dx}$. The general solution is then: $latex y \cdot \text{I.F.} = \int Q(x) \cdot \text{I.F.} , dx + C$ where $latex C$ is the constant of integration.
Applications
First-order differential equations are widely used to model simple physical systems and processes, such as exponential growth and decay, cooling and heating processes (Newton’s Law of Cooling), and the motion of objects with constant acceleration. These equations provide insight into how quantities change over time under various conditions.
Geometric Interpretation
The solution to a first-order differential equation can be interpreted geometrically as the family of curves in the $latex xy$-plane that satisfy the equation. Each curve in the family is a solution corresponding to a particular value of the integration constant $latex C$.
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