Question

Solving a First-Order Linear Differential Equation In Exercises \(51-58\) , solve the first-order differential equation by any appropriate method. $$ x d x+\left(y+e^{y}\right)\left(x^{2}+1\right) d y=0 $$

Short answer

The solution to the differential equation can be found by identifying M and N, checking if the equation is exact, solving the exact differential equation, and then determining the solution of the equation by equating Φ(x, y) to a constant C.

Step-by-step solution

Identify M and N

Separate the given differential equation into two parts M(x, y) dx and N(x, y) dy. Here, it is evident that M(x, y) = x and N(x, y) = (y + e^y)(x^2 + 1)

Check if the equation is exact.

We know that a differential equation is exact if and only if ∂M/∂y = ∂N/∂x. Compute these partial derivatives. If they are equal, the equation is exact; if not, a integrating factor needs to be found to make the equation exact.

Solve the Exact Differential Equation

Find the potential function Φ(x, y) whose gradient is [M(x,y), N(x,y)]. This is achieved by integrating M(x, y) with respect to x and N(x, y) with respect to y, and then equating the two resulting expressions and solving for the arbitrary function.

Determine the solution of the differential equation

After finding Φ(x, y), the solution of the differential equation is defined by Φ(x, y) = C.

Key concepts

First-Order Differential Equations

First-order differential equations are equations that involve the first derivative of a function. They are fundamental in mathematical modeling because they describe a wide range of real-world phenomena. The general form of a first-order differential equation is given as \( \frac{dy}{dx} = f(x, y) \), where \( f(x, y) \) is a function of \( x \) and \( y \).
These equations can be classified further into linear and non-linear equations, and knowing which type you're dealing with is crucial for finding the right solution method.

• In linear first-order differential equations, the terms involve derivatives with constant coefficients or functions not multiplied by the dependent variable or its derivative.
• In non-linear equations, at least one term includes the dependent variable or its derivative raised to a power or inside another function.

Understanding the distinction between these kinds of differential equations helps recognize what techniques to employ in arriving at a solution.

Exact Equations

Exact differential equations are a special type of first-order differential equations. They can be solved using integration directly, thanks to a relationship between their components, which is checked using partial derivatives.
The differential equation \( M(x, y)dx + N(x, y)dy = 0 \) is exact if the partial derivative of \( M \) with respect to \( y \), \( \frac{\partial M}{\partial y} \), is equal to the partial derivative of \( N \) with respect to \( x \), \( \frac{\partial N}{\partial x} \).
When the equation is exact, there exists a potential function \( \Phi(x, y) \) such that the gradient of this function gives us \( M \) and \( N \).

• This means \( \Phi_x(x, y) = M(x, y) \)
• And \( \Phi_y(x, y) = N(x, y) \)

If the equation isn't exact, an integrating factor may convert it into an exact equation. This provides a way to handle a broader range of complex differential equations.

Partial Derivatives

Partial derivatives are crucial when dealing with functions of multiple variables. They represent the derivative of a function with respect to one of the variables, holding the others constant.
In the context of differential equations, partial derivatives help determine whether a given equation is exact. For example, in confirming exactness, you check if \( \frac{\partial M}{\partial y} \) equals \( \frac{\partial N}{\partial x} \).
Partial derivatives are foundational because they directly relate to the rate of change of the function in relation to each of its variables independently. This provides insight into how systems behave and interact with each variable affecting the outcome separately.
Mastering partial derivatives helps in advanced calculus and is necessary for understanding gradients, which are used frequently in vector calculus and differential equations.

Integrating Factor

An integrating factor is a function that, when multiplied by a non-exact differential equation, transforms it into an exact equation. This powerful technique simplifies solving differential equations that initially seem complex.
The process of finding an integrating factor involves identifying a function \( \mu(x, y) \) that makes the equation exact. The concept behind using an integrating factor is to balance the equation so that the partial derivatives align as required by exact equations.

• Once found, multiply the entire differential equation by this integrating factor.
• Then proceed by solving the transformed equation as an exact differential equation.

This method bridges the gap where the equation's structure doesn't allow for direct integration and extends the capability to solve a wider range of equations analytically.