Question

a. Given the equation \(f^{\prime}+a f=b,(a \neq 0),\) make the substitution \(f(x)=g(x)+b / a\) and obtain a differential equation for \(g\). Then derive the general solution for \(f^{\prime}+a f=b\). b. Find the general solution to \(f^{\prime}+f=2\).

Short answer

The general solution is \( f(x) = C e^{-x} + 2 \).

Step-by-step solution

Substitute and Simplify

Given the differential equation \( f' + a f = b \), substitute \( f(x) = g(x) + \frac{b}{a} \) into the equation. This gives us: \( (g(x) + \frac{b}{a})' + a (g(x) + \frac{b}{a}) = b \).

Differentiate and Rearrange Terms

Differentiate the substitution: \( g'(x) + 0 \). So the modified equation becomes \( g'(x) + a g(x) + \frac{b}{a}a = b \). Simplify to get: \( g'(x) + a g(x) = b - b = 0 \).

Solve the Homogeneous Equation

Solve \( g'(x) + a g(x) = 0 \). Recognize this as a first-order linear homogeneous differential equation. The solution is \( g(x) = C e^{-ax} \) where \( C \) is a constant.

Find General Solution for f

Substitute back to find \( f(x) \): \( f(x) = g(x) + \frac{b}{a} = C e^{-ax} + \frac{b}{a} \). This is the general solution for the original differential equation \( f' + af = b \).

Apply to Specific Case

For \( f' + f = 2 \), set \( a = 1 \) and \( b = 2 \). Use the general form \( f(x) = C e^{-x} + \frac{2}{1} \).

Simplify the Particular Solution

For the specific equation, the particular solution simplifies to: \( f(x) = C e^{-x} + 2 \). This is the general solution.

Key concepts

First-order Linear Differential Equation

First-order linear differential equations are fundamental in understanding how functions change with respect to another variable. These equations are written in the form \( f'(x) + a f(x) = b \), where \( f'(x) \) denotes the derivative of \( f \) with respect to \( x \), and \( a \) and \( b \) are constants. These equations describe situations where the rate of change of a quantity is proportional to the quantity itself, modified by a linear term.

In practice, these equations often arise in a range of fields such as physics, engineering, and economics. A common approach to solve them involves finding both a homogeneous and a particular solution, which are then combined to form the general solution. Understanding the structure of first-order linear differential equations unlocks the ability to analyze systems with constant rates of change and external influences.

Homogeneous Solution

The homogeneous solution is an essential component when solving differential equations. It addresses the situation described by the homogeneous version of the equation, where the non-homogeneous term is set to zero. For a first-order linear differential equation of the form \( f' + a f = 0 \), the homogeneous solution explores the behavior of the system without external forces.

The solution to this is typically an exponential function, \( g(x) = C e^{-ax} \), where \( C \) is an integration constant that can be determined if initial conditions are provided. This exponential behavior reflects scenarios like exponential decay or growth, dependent on the sign of \( a \). Homogeneous solutions are crucial for understanding the natural behavior of systems.

Particular Solution

The particular solution of a differential equation specifically satisfies the equation as given, including the non-homogeneous term. It represents a steady-state or a specific response of the system under the influence of an external factor—in this case, the constant \( b \).

For a given first-order linear equation, once a homogeneous solution is obtained, the particular solution can be found by substituting and simplifying, as in the original substitution given with \( f(x) = g(x) + \frac{b}{a} \). This particular solution compensates for the effect of the constant \( b \) added to the system, allowing us to see how the system behaves over time, eventually combining with the homogeneous solution for an all-encompassing general solution.

General Solution

The general solution of a differential equation is the comprehensive form that encompasses every possible solution. It combines the homogeneous and particular solutions, providing a full description of the system's behavior.

For our equation \( f'(x) + a f(x) = b \), the general solution is formulated as \( f(x) = C e^{-ax} + \frac{b}{a} \). Here, the term \( C e^{-ax} \) represents the homogeneous solution, describing intrinsic system behaviors, while \( \frac{b}{a} \) accounts for the constant external influence.

• This combination ensures that all initial or boundary conditions applicable to the problem can be satisfied.
• It allows us to model both the natural and induced behavior of dynamic systems efficiently.

Understanding how to find and interpret the general solution is vital for predicting how real-world systems evolve over time.