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Chapter 2: Problem 21

Obtain the solution of the differential equation \\[\dot{x}+a x=A \sin \omega t, \quad x(0)=b\\]

Short Answer

Expert verified
The solution is \( x(t) = \left( b + \frac{A \omega}{ a^2 + \omega^2 } \right)e^{-at} + \frac{aA}{ a^2 + \omega^2 } \sin(\omega t) - \frac{A \omega}{ a^2 + \omega^2 } \cos(\omega t) \).

Step by step solution

01

Identify the type of differential equation

The given equation is a first-order linear non-homogeneous differential equation:\[ \frac{dx}{dt} + a x = A \sin \omega t. \]
02

Solve the corresponding homogeneous equation

First, solve the homogeneous equation:\[ \frac{dx}{dt} + a x = 0 \] To solve this, separate the variables and integrate: \[ \frac{dx}{x} = -a dt \] Integrate both sides: \[ \ln{|x|} = -at + C \] Therefore, the solution to the homogeneous equation is: \[ x_h(t) = C e^{-at} \]
03

Find the particular solution

Next, find a particular solution to the non-homogeneous equation. Try a solution of the form: \[ x_p(t) = B \sin(\omega t) + C \cos(\omega t). \] Substituting \( x_p(t) \) into the non-homogeneous equation, determine the coefficients \( B \) and \( C \): \[ \dot{x}_p + ax_p = A \sin(\omega t). \]
04

Substitute the particular solution and solve for coefficients

First, compute the derivative \( \dot{x}_p \): \[ \dot{x}_p(t) = B \omega \cos(\omega t) - C \omega \sin(\omega t). \] Substitute \( x_p(t) \) and \( \dot{x}_p(t) \) into the differential equation: \[ B \omega \cos(\omega t) - C \omega \sin(\omega t) + a (B \sin(\omega t) + C \cos(\omega t)) = A \sin(\omega t). \] Equate coefficients of \( \cos(\omega t) \) and \( \sin(\omega t) \) to get: \[ B \omega + aC = 0 \] and \[ -C \omega + aB = A. \]
05

Solve for coefficients B and C

From the equations: \[ B \omega + aC = 0 \implies B = -\frac{aC}{\omega} \]\[ -C \omega + a(-\frac{aC}{\omega}) = A \implies -C \omega - a^2\frac{C}{\omega} = A \]\[ C ( - \omega - \frac{a^2}{\omega} ) = A \] Therefore, \[ C = - \frac{A \omega}{ a^2 + \omega^2 } \] and \[ B = -\frac{aC}{\omega} = \frac{aA}{ a^2 + \omega^2 } \]
06

Write the general solution

The general solution is the sum of the homogeneous and particular solutions: \[ x(t) = x_h(t) + x_p(t) = C_1 e^{-at} + \frac{aA}{ a^2 + \omega^2 } \sin(\omega t) - \frac{A \omega}{ a^2 + \omega^2 } \cos(\omega t). \]
07

Apply the initial condition to find the constant

Use the initial condition \( x(0) = b \) to find \( C_1 \): \[ b = C_1 + \frac{aA}{ a^2 + \omega^2 } \sin(0) - \frac{A \omega}{ a^2 + \omega^2 } \cos(0) \implies C_1 = b + \frac{A \omega}{ a^2 + \omega^2 }. \]
08

Write the final solution

Substitute \( C_1 \) back into the general solution: \[ x(t) = \left( b + \frac{A \omega}{ a^2 + \omega^2 } \right)e^{-at} + \frac{aA}{ a^2 + \omega^2 } \sin(\omega t) - \frac{A \omega}{ a^2 + \omega^2 } \cos(\omega t). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Linear Differential Equation
A differential equation describes a relationship involving a function and its derivatives. If an equation involves the first derivative only, it is termed a first-order differential equation. The standard form of a first-order linear differential equation is:
\[ \frac{dy}{dx} + P(x) y = Q(x) \tag{1} \]
where \( P(x) \) and \( Q(x) \) are given functions.
Key Characteristics:
  • The equation is 'linear' if the function and its derivative are to the first power and not multiplied with each other.
  • It is 'first-order' because it involves the first derivative \( \frac{dy}{dx} \)
  • The equation can be homogeneous or non-homogeneous based on the form of \( Q(x) \).
Applying this knowledge to our exercise:
The given equation: \( \frac{dx}{dt} + a x = A \text{sin}(\text{蠅} t) \) fits the form of a first-order linear differential equation with \( P(t) = a \) and \( Q(t) = A \text{sin}(\text{蠅} t) \). This identification helps in choosing the right methods for solving it.
Non-Homogeneous Differential Equation
A first-order differential equation can be classified as homogeneous or non-homogeneous depending on the function \( Q(x) \).
Understanding Homogeneous vs. Non-Homogeneous:
  • If \( Q(x) = 0 \), the equation is termed homogeneous.
  • If \( Q(x) \) is not equal to zero, the equation is considered non-homogeneous.
The differential equation in the problem: \( \dot{x} + a x = A sin(\omega t) \) is non-homogeneous because \( A \sin(\omega t) \) is not zero. To solve a non-homogeneous differential equation:
  1. First, solve the homogeneous portion of the equation (i.e., \( \dot{x} + a x = 0 \)).
  2. Next, find a 'particular solution' for the non-homogeneous part (i.e., \( A \sin(\omega t) \)).
Combining these solutions gives the general solution to the non-homogeneous equation.
Particular Solution
To solve a non-homogeneous differential equation, we need to find a specific solution that satisfies the equation.
Finding the Particular Solution:
This involves determining a form of \( x_p(t) \) that turns the equation \( \dot{x} + a x = A \text{sin}(\text{\text{蠅}} t) \) into a true statement. We assume a form: \( x_p(t) = B \text{sin}(\text{蠅} t) + C \text{cos}(\text{蠅} t) \).

Next, differentiate to find \( \text{\text{dot{x}}}_p \):
\[ \text{\text{dot{x}}}_p(t) = B \text{\text{蠅}} \text{cos}(\text{\text{蠅}} t) - C \text{\text{蠅}} \text{sin}(\text{\text{蠅}} t). \] Substituting both \( x_p(t) \) and \( \text{\text{dot{x}}}_p(t) \) back into the non-homogeneous equation, we solve for coefficients \( B \) and \( C \):
  • Combining algebraic terms helps isolate these coefficients.
  • The end result will show: \[ C = - \frac{A \text{蠅}}{a^2 + \text{蠅}^2 } \] and \[ B = \frac{aA}{a^2 + \text{蠅}^2 } \]

These values inserted back into\( x_p(t) \) form the particular solution needed.

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Most popular questions from this chapter

Solve the differential equation \\[\dot{x}+2 x=\delta(t), \quad x(0-)=0\\]

Find the Laplace transforms of the following functions: (a) \(f_{1}(t)=0, \quad\) for \(t<0\) \(=e^{-0.4 i} \cos 12 t, \quad\) for \(t \geq 0\) (b) \(f_{2}(t)=0, \quad\) for \(t<0\) \(=\sin \left(4 t+\frac{\pi}{3}\right), \quad\) for \(t \geq 0\)

Find the Laplace transforms of the following functions: (a) \(f_{1}(t)=0, \quad\) for \(t<0\) \(=3 \sin \left(5 t+45^{\circ}\right), \quad\) for \(t \geq 0\) (b) \(f_{2}(t)=0, \quad\) for \(t<0\) \(=0.03(1-\cos 2 t), \quad\) for \(t \geq 0\)

Find the inverse Laplace transform of \\[F(s)=\frac{s+1}{s\left(s^{2}+s+1\right)}\\]

Obtain the partial-fraction expansion of the following function with \\[F(s)=\frac{10(s+2)(s+4)}{(s+1)(s+3)(s+5)^{2}}\\] Then, obtain the inverse Laplace transform of \(F(s)\).
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