Problem 22 - Calculus and its applications

Question

(a) find the general solution of each differential equation, and (b) check the solution by substituting into the differential equation. \(\frac{d Q}{d t}=2 Q\)

Step-by-Step Solution

Answer

General solution: \( Q(t) = Ce^{2t} \). Verified by substitution.

Step 1: Identify the Type of Differential Equation

The differential equation given is \( \frac{dQ}{dt} = 2Q \). This is a first-order linear ordinary differential equation (ODE). Its standard form is \( \frac{dy}{dt} = ky \), where \( k \) is a constant. In this equation, \( k = 2 \).

Step 2: Solve the Differential Equation

For a first-order linear ODE in the standard form \( \frac{dy}{dt} = ky \), the solution is \( y(t) = Ce^{kt} \), where \( C \) is an arbitrary constant. Therefore, the solution to our equation \( \frac{dQ}{dt} = 2Q \) is: \[ Q(t) = Ce^{2t} \] where \( C \) is an arbitrary constant that can be determined by an initial condition if given.

Step 3: Substitute the Solution into the Original Differential Equation

To verify the solution, substitute \( Q(t) = Ce^{2t} \) back into the original differential equation. We need to differentiate \( Q(t) \):\[ \frac{d}{dt} [Ce^{2t}] = C \cdot 2e^{2t} = 2Ce^{2t} \] The right side of the original equation \( \frac{dQ}{dt} = 2Q \) is \( 2Q = 2(Ce^{2t}) = 2Ce^{2t} \). Therefore, the left-hand side equals the right-hand side, confirming that the solution is correct.

Key Concepts

First-Order Linear ODE

A first-order linear ordinary differential equation (ODE) is one of the most fundamental types of differential equations. It involves a derivative of the first degree and can be expressed in the form \( \frac{dy}{dt} = ky \). Here, \( k \) is a constant which influences the rate of change of \( y \) with respect to \( t \). Such equations often model exponential growth or decay processes in natural systems.

Understanding the structure of these ODEs is crucial. The 'linear' aspect means that the function \( y \) and its derivative \( \frac{dy}{dt} \) are not multiplied together, and the equation relies on the superposition principle. This makes these equations much easier to solve and analyze compared to nonlinear differential equations.

Recognizing the first-order linear ODE in a problem allows you to apply standardized techniques to find solutions, simplifying complex real-world problems. In our case, \( \frac{dQ}{dt} = 2Q \), we identify \( k = 2 \), which directly informs the nature of the solution.

General Solution

The general solution for a first-order linear ODE is typically exponential in nature. Once the equation is rewritten in its standard form \( \frac{dy}{dt} = ky \), finding the solution is straightforward. The solution is given by \( y(t) = Ce^{kt} \), where \( C \) is an arbitrary constant.

• \( C \): Represents the initial condition or starting value. It is crucial because it adjusts the general solution to fit specific initial conditions given in real situations.
• \( e^{kt} \): Shows exponential behavior dictated by \( k \). A positive \( k \) indicates growth, whereas a negative \( k \) represents decay.

In the example \( \frac{dQ}{dt} = 2Q \), the solution becomes \( Q(t) = Ce^{2t} \). This reveals how \( Q(t) \) grows exponentially with respect to time, depending on both the initial amount \( C \) and the growth rate \( k = 2 \). Understanding this general form equips you to tackle a wide array of similar problems in various fields like physics, biology, and finance.

Verification of Solution

Verification involves substituting the proposed solution back into the original differential equation to confirm its validity. For the given equation, the solution \( Q(t) = Ce^{2t} \) needs to satisfy the original equation \( \frac{dQ}{dt} = 2Q \).

Whenever you solve a differential equation, verifying the solution is vital to ensure correctness. Here's how you do it:

• Differentiate your solution \( Q(t) = Ce^{2t} \) to find \( \frac{dQ}{dt} \). This is \( C \cdot 2e^{2t} \).
• Compare \( \frac{dQ}{dt} \) to \( 2Q \). Substituting, \( 2Q = 2(Ce^{2t}) = 2Ce^{2t} \).

The left-hand side equals the right-hand side, confirming our proposed solution is correct. This agreement verifies that \( Q(t) = Ce^{2t} \) accurately models the behavior specified by the differential equation, ensuring you have calculated your solution correctly and reliably.