Find a particular solution for the given differential equation. d²q dq La +Ra + q = Vosin (wt) dt dt

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Problem Statement:**

Find a particular solution for the given differential equation.

**Differential Equation:**

\[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = V_0 \sin (\omega t) \]

---

**Explanation:**

This equation represents a second-order linear differential equation with constant coefficients, often used to model electrical circuits involving inductance (\(L\)), resistance (\(R\)), and capacitance (\(C\)). The term \(V_0 \sin (\omega t)\) on the right side represents an external sinusoidal driving force. The goal is to find a particular solution \(q(t)\) that satisfies this equation.
Transcribed Image Text:**Problem Statement:** Find a particular solution for the given differential equation. **Differential Equation:** \[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = V_0 \sin (\omega t) \] --- **Explanation:** This equation represents a second-order linear differential equation with constant coefficients, often used to model electrical circuits involving inductance (\(L\)), resistance (\(R\)), and capacitance (\(C\)). The term \(V_0 \sin (\omega t)\) on the right side represents an external sinusoidal driving force. The goal is to find a particular solution \(q(t)\) that satisfies this equation.
Expert Solution
Step 1: Use the formula f(D^2)y =sinax

We will use the following formula to get the particular solution of the above differential equation

                         

                                                 f(D2)y=sin(ax)y=1f(D2)sin(ax)

                              

                                                          y=1f(a2)sin(ax) ; provided f(a2)0

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