in finding the particular solution, how do you calculate the derivaties?

When finding the particular solution of a differential equation, we often assume a form for the solution based on the type of non-homogeneous term. After assuming the form, we need to calculate its derivatives to substitute them back into the original differential equation.

Let's revisit the example we discussed:
Given the equation:
\[
\frac{d^2y}{dx^2} + 7\frac{dy}{dx} + 12y = e^{3x}
\]

We assumed a particular solution of the form:
\[
y_p(x) = Ae^{3x}
\]
where \(A\) is a constant to be determined.

### Calculating Derivatives
1. **First Derivative**:
   \[
   \frac{dy_p}{dx} = \frac{d}{dx}(Ae^{3x})
   \]
   Here, \(A\) is a constant and \(e^{3x}\) is an exponential function. To differentiate \(e^{3x}\), we use the chain rule. The derivative of \(e^{3x}\) with respect to \(x\) is:
   \[
   \frac{d}{dx}(e^{3x}) = 3e^{3x}
   \]
   Therefore:
   \[
   \frac{dy_p}{dx} = 3Ae^{3x}
   \]

2. **Second Derivative**:
   Now we calculate the second derivative:
   \[
   \frac{d^2y_p}{dx^2} = \frac{d}{dx}\left(3Ae^{3x}\right)
   \]
   Again, \(3A\) is a constant, and the derivative of \(e^{3x}\) with respect to \(x\) is still \(3e^{3x}\). So:
   \[
   \frac{d^2y_p}{dx^2} = 3 \times 3Ae^{3x} = 9Ae^{3x}
   \]

### Summary of Derivatives
- The **first derivative** of \(y_p(x) = Ae^{3x}\) is:
  \[
  \frac{dy_p}{dx} = 3Ae^{3x}
  \]
- The **second derivative** of \(y_p(x)\) is:
  \[
  \frac{d^2y_p}{dx^2} = 9Ae^{3x}
  \]

These derivatives are then substituted back into the original differential equation to solve for the constant \(A\).

### General Method for Derivatives
- **Exponential functions**: For a function like \(e^{kx}\), the derivative is \(ke^{kx}\).
- **Polynomial functions**: For a function like \(ax^n\), the derivative is \(nax^{n-1}\).
- **Trigonometric functions**: For a function like \(\sin(kx)\) or \(\cos(kx)\), the derivative would be \(k\cos(kx)\) or \(-k\sin(kx)\), respectively.

These basic rules are applied repeatedly depending on the assumed form of the particular solution.

Understand how to calculate derivatives for finding particular solutions of differential equations. Learn step-by-step with an example differential equation involving an exponential non-homogeneous term. Suitable for college-level students and anyone studying differential equations.

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