Question

Are the following differential equations linear? Explain your reasoning. $$ y^{\prime}=y+e^{y} $$

Short answer

The equation is nonlinear due to the presence of the exponential term \( e^{y} \).

Step-by-step solution

Identify the Terms of the Equation

First, we need to identify all the terms present in the given differential equation, which is \( y^{\prime} = y + e^{y} \). Here, \( y^{\prime} \) is the derivative of \( y \) with respect to \( x \), and the terms on the right are \( y \) and \( e^{y} \).

Check for Linearity Criteria

A differential equation is linear if it can be expressed in the form \( a_1(x) y^{(n)} + a_2(x) y^{(n-1)} + \,\ldots\, + a_n(x)y = g(x) \), where the coefficients \( a_i(x) \) and \( g(x) \) are functions of \( x \) alone, and \( y \) and its derivatives are all of the first degree. Specifically, there should be no terms that involve \( y \) raised to a power other than one, or composed inside a nonlinear function like an exponential, sine, cosine, etc.

Identify Nonlinearity

In our equation, \( y \) appears both linearly as \( y \) and non-linearly as \( e^{y} \). The term \( e^{y} \) is an exponential function involving \( y \), which makes the equation nonlinear. For linearity, any transformations on \( y \) must be limited to simple multiplication by functions of \( x \), which is not the case here.

Conclusion: Determine Linearity of the Equation

Since the equation \( y^{\prime} = y + e^{y} \) contains an exponential \( e^{y} \), it does not meet the criteria for a linear differential equation, which requires all terms involving \( y \) and its derivatives to not be in nonlinear functions.

Key concepts

Linearity

Understanding linearity in differential equations is crucial for solving them effectively. A differential equation is termed "linear" if each term is either a constant or a product of a constant and the first power of the dependent variable or its derivatives. Here, linearity allows the equation to conform to a simpler form, making it easier to apply analytical solutions or certain numerical techniques. For example, in the linear form \[a_1(x) y^{(n)} + a_2(x) y^{(n-1)} + \.\.\. + a_n(x)y = g(x), \] the coefficients \(a_i(x)\) and the function \(g(x)\) depend entirely on the independent variable, \(x\), rather than the dependent variable, \(y\).
A linear equation does not involve powers or functions of \(y\) like squaring, square roots, or exponentials. However, keep in mind that all variables involved must be treated as first-degree, free from any higher powers or combinations that could complicate the system.

• Each term in the equation must relate directly to \(y\) or its derivatives, linearly.
• The equation should have no products involving \(y\) or its derivatives multiplying together.
• Nonlinearity in any term deviates the equation from the linear form.

Nonlinear Terms

Nonlinear terms in differential equations introduce complexities because they involve powers or functions of a dependent variable or its derivatives beyond the first degree. These complexities can change the behavior or solutions of the equation. For instance, in the differential equation \( y' = y + e^y \), the presence of the exponential term \( e^y \) introduces nonlinearity.
The effects of nonlinear terms are significant:

• They make the equation more difficult to solve analytically because standard methods for linear equations often don’t apply.
• Nonlinear terms can result in multiple solutions, unique dynamics, or no solutions at all, depending on the form of the equation.
• They often necessitate the use of numerical techniques or approximations to understand the solutions.

When assessing a differential equation, identifying such terms is essential, as they dictate the types of methods and tools that can be used to solve the equation.

Exponential Functions

Exponential functions are mathematically represented as \(e^x\) or \(a^x\), where \(e\) is the Euler's number and \(a\) represents any constant. These functions are important in various applications but introduce nonlinearity when they appear in differential equations. For instance, in the equation \( y' = y + e^y \), the term \( e^y \) affects how the solutions behave dramatically.
Exponential functions in equations:

• Create nonlinearity by making it impossible to express the relationship as a simple sum of first-degree terms of \(y\) and its derivatives.
• Introduce dynamic behaviors in solutions, often leading to explosive growth or decay in variables over time.
• Need special consideration when modeling real-world processes, as they often represent phenomena like population growth, radioactive decay, and more.

Any appearance of an exponential function in a differential equation should be carefully assessed for its impact on the solution strategy and the understanding of the problem's nature.

Criteria for Linearity

For a differential equation to be classified as linear, it must satisfy specific criteria. Recognizing these criteria helps differentiate between solvable cases using standard linear approaches and those requiring advanced methods.

Key Criteria:

1. The equation should allow representation in the form
\[a_1(x) y^{(n)} + a_2(x) y^{(n-1)} +...+ a_n(x)y = g(x)\]
where all coefficients \(a_i(x)\) and \(g(x)\) depend only on the independent variable \(x\).
2. There must be no products, powers, or compositions involving \(y\) or its derivatives, such as \(y^2\), \(e^y\), or \(\sin(y)\).
3. Terms must be of the first degree—no variable expressions raised to powers other than one.
4. Any derivatives involved must also appear linearly, making the equation more straightforward to solve.
When examining an equation, these criteria serve as a checklist to determine the appropriate methods of solution, whether analytical or numerical.