Differential Equations Made Easy: Solutions and Notes Exercise 15.1 for class 12

Hi everyone! welcome back another video today i am going to reveal notes of differential equation Exercise 15.1 let's ready to levels up your knowledge
What is a Differential Equation?
A *differential equation* is an equation that relates a function to its derivatives. In simple words, it is an equation that involves an unknown function and its rates of change. Here’s how you can think of it:
**Function**: A function is like a rule that takes an input (like time or distance) and gives you an output (like speed or position).
**Derivative**: The derivative of a function tells you how the function is changing at any point; it's like the slope of the function at that point.
Example
If you have a function \( y = f(x) \), the first derivative of that function, written as \( f'(x) \) or \( \frac{dy}{dx} \), tells you how fast \( y \) is changing with respect to \( x \). A simple differential equation could look like this:
\[
\frac{dy}{dx} = 3y
\]
This means that the rate of change of \( y \) with respect to \( x \) is proportional to \( y \) itself.
Types of Differential Equations
1. **Ordinary Differential Equations (ODEs)**: These involve functions of a single variable and their derivatives. For example, \(\frac{dy}{dx} + y = 0\).
2. **Partial Differential Equations (PDEs)**: These involve functions of multiple variables and their partial derivatives. An example would be the heat equation in physics.
General vs. Particular Solutions
A *general solution* contains constants (like \( C \)) and represents a family of solutions.
A *particular solution* is obtained from the general solution by using initial or boundary conditions.
Important Formulas and Techniques
1. **Separation of Variables**: This method is used when you can separate the variables on opposite sides of the equation. For example:
Given \(\frac{dy}{dx} = g(y)h(x)\), you can rearrange it to:
\[
\frac{dy}{g(y)} = h(x)dx
\]
Then, integrate both sides.
2. **Integrating Factor**: For a first-order linear differential equation of the form:
\[
\frac{dy}{dx} + P(x)y = Q(x)
\]
The integrating factor \( \mu(x) = e^{\int P(x) dx} \) helps in simplifying the equation to solve it.
3. **Homogeneous Equation**: If you have an equation \( \frac{dy}{dx} = f\left(\frac{y}{x}
ight) \), it can often be simplified using substitution \( v = \frac{y}{x} \).
Conclusion
Differential equations are a powerful tool in mathematics, helping us model real-world situations like growth, decay, and motion. The key steps are identifying the type of differential equation you have and choosing the appropriate method to solve it.
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Differential Equations Made Easy: Solutions and Notes for Class 12, Exercise 15.1