Question

Show that if \(\left(N_{x}-M_{y}\right) /(x M-y N)=R,\) where \(R\) depends on the quantity \(x y\) only, then the differential equation $$ M+N y^{\prime}=0 $$ has an integrating factor of the form \(\mu(x y)\). Find a general formula for this integrating factor.

Short answer

#Short Answer# The integrating factor for the given differential equation can be found using the given relation and it has the general form of \(\mu(xy) = ke^{R(xy)}\), where \(k\) is a constant.

Step-by-step solution

Compute the total differential of the given relation

Given that \(\frac{N_x - M_y}{xM - yN} = R\) where \(R\) depends on \(xy\) only. Let's find the total differential of both sides with respect to \(x\) and \(y\): $$d\left(\frac{N_x - M_y}{xM - yN}\right) = dR$$

Equating the total differentials

From \(d\left(\frac{N_x - M_y}{xM - yN}\right) = dR\), we know that \(R\) depends only on \(xy\). Therefore, we can write \(dR = R_xdx + R_ydy\). Equating the total differentials, we get: $$d\left(\frac{N_x - M_y}{xM - yN}\right) = R_xdx + R_ydy$$

Rewrite the given differential equation

The given differential equation is \(M + Ny' = 0\). We need to rewrite this equation using the total differential notation: $$Mdx + Ndy = 0$$

Find the integrating factor

Now, we'll try to find the integrating factor in the form of \(\mu(xy)\). An integrating factor \(\mu\) satisfies the condition: $$\frac{d\mu}{\mu} = R_xdx + R_ydy$$ We know that \(R = R(xy)\), so we can write: $$\frac{d\mu}{\mu} = R_xdx + R_ydy \Rightarrow \frac{d\mu}{\mu} = (R_xdx + R_ydy) = dR$$ Integrating both sides with respect to the appropriate variable, we get: $$\int\frac{d\mu}{\mu} = \int dR \Rightarrow \ln\mu = R(xy) + C$$ Where \(C\) is the integration constant. Now, taking the exponent of both sides, we get the integrating factor \(\mu(xy)\): $$\mu(xy) = e^{R(xy) + C} = e^Ce^{R(xy)}$$ Since \(e^C\) is also a constant, we can write the integrating factor as: $$\mu(xy) = ke^{R(xy)}$$ Where \(k\) is a constant. Thus, we've found the integrating factor in the form \(\mu(xy)\).

Key concepts

Differential Equation

A differential equation is a mathematical equation that involves functions and their derivatives. These equations are essential in modeling real-world phenomena and understanding the behavior of dynamic systems. For example, they can model how heat spreads through a solid, how populations grow over time, or how electrical currents vary in circuits.

In simple terms, a differential equation relates a function with its rate of change. Consider the equation given in the original problem: \[M + Ny' = 0\]Here, \(M\) and \(N\) are functions of certain variables, and \(y'\) represents the derivative of \(y\) with respect to another variable, usually \(x\). This shows how changes in \(y\) depend on the relationship between \(M\) and \(N\).

Solving a differential equation often means finding the function \(y\) that satisfies this relationship under certain conditions. Integrating factors can be used to simplify the process, especially in equations that are not easily separable or integratable directly.

Total Differential

The total differential of a function expresses how the function changes when the variables it depends on change. It's a crucial tool in multivariable calculus, especially for functions of two or more variables.

When you have a relationship given by \(f(x, y)\), the total differential \(df\) tells us how small changes \(dx\) and \(dy\) in \(x\) and \(y\), respectively, affect the function's value. The formula for total differential is:\[df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\]In the original solution, this concept is used to equate the changes in both sides of the given relation, \(d\left(\frac{N_x - M_y}{xM - yN}\right) = dR\).

This equality ensures that if changes in one part of the equation are expressed correctly, they match the changes in another part, which is essential for solving such problems.

R(xy)

In the problem setup, \(R(xy)\) is a crucial function that depends solely on the product of variables \(x\) and \(y\). Recognizing functions based on combined variables, like \(xy\), can simplify the solving process of differential equations significantly.

This form is pivotal because it allows us to identify an integrating factor \(\mu(xy)\) in terms of this function. We find the integrating factor as follows:\[\mu(xy) = ke^{R(xy)}\]where \(k\) is a constant. This form enables us to transform the differential equation into an easily integratable form.

Thus, recognizing functions like \(R(xy)\) helps streamline the process of solving such complex equations by reducing the variables and finding consistent factors that simplify the original problem. This insight is often key for students trying to find solutions to seemingly complicated mathematical problems.