Question

Solve the initial-value problems. Show that if the equation \(M(x, y) d x+N(x, y) d y=0\) is both homogeneous and exact and if \(M(x, y) x+N(x, y) y\) is not a constant, then the solution of this equation is \(M(x, y) x+N(x, y) y=c\), where \(c\) is an arbitrary constant.

Short answer

We are given a homogeneous and exact differential equation, \(M(x, y) \, dx + N(x, y) \, dy = 0\), with the condition that \(M(x, y)x + N(x, y)y\) is not constant. After verifying homogeneity and exactness and considering the given condition, we find the solution in the desired form as: \[M(x, y)x + N(x, y)y = c\] where \(c\) is an arbitrary constant.

Step-by-step solution

Review definitions of homogeneous and exact equations

Recall that an equation is homogeneous if it can be written in the form \(M(tx, ty) = t^nM(x, y)\) and \(N(tx, ty) = t^nN(x, y)\) for some integer \(n\). An equation is exact if there exists a function \(F(x, y)\), such that \(M(x, y) = \frac{\partial F}{\partial x}\) and \(N(x, y) = \frac{\partial F}{\partial y}\).

Verify homogeneity

We are given that the equation is homogeneous. So we have: \[M(tx, ty) = t^nM(x, y)\] \[N(tx, ty) = t^nN(x, y)\] for some integer \(n\).

Verify exactness

We are also given that the equation is exact. Using the exactness definition, we have: \[M(x, y) = \frac{\partial F}{\partial x}\] \[N(x, y) = \frac{\partial F}{\partial y}\]

Check the condition \(M(x, y)x + N(x, y)y\neq constant\)

The condition states that \(M(x, y)x + N(x, y)y\) is not constant. It means that \(M(x, y)x + N(x, y)y\) should have some dependence on \(x\) and/or \(y\).

Find the solution in terms of M(x, y) and N(x, y)

Since we have verified homogeneity and exactness, and we are given that \(M(x, y)x + N(x, y)y\) is not constant, we can now find the solution in terms of \(M(x, y)\) and \(N(x, y)\). Taking the expression, and arranging terms to obtain the solution: \[M(x, y)x + N(x, y)y = c\] Where \(c\) is an arbitrary constant. This is our desired solution form.

Key concepts

Homogeneous Equations

A homogeneous equation is a type of differential equation that maintains a consistent structure when scaling both the variables and the functions. For an equation to be considered homogeneous, the functions involved should satisfy the properties:

• For function \(M(x, y)\): \(M(tx, ty) = t^nM(x, y)\)
• For function \(N(x, y)\): \(N(tx, ty) = t^nN(x, y)\)

Here, "\(t\)" represents a scalar factor, and \(n\) is the degree of homogeneity. This simply means if you scale your variables, the entire equation will scale in a uniform way by a factor of \(t^n\).
Homogeneous equations often appear in physics and engineering problems where systems behave proportionally, such as problems involving waves or heat conduction. Understanding how these systems scale aids in simplifying complex problems by revealing symmetry or invariance properties.
In the context of an exact differential equation, proving that \(M(x, y)\) and \(N(x, y)\) are homogeneous is often the first step before exploring further solutions and properties of the equation.

Initial-Value Problems

In the realm of differential equations, an Initial-Value Problem (IVP) involves finding a particular solution that not only satisfies the differential equation but also meets specific conditions at a given point. These conditions are outlined at the initial stage of the function or trajectory, hence the name "initial-value."
An IVP typically comes in the form; for instance, if we have a differential equation \(dy/dx = g(x, y)\), and we know \(y(x_0) = y_0\), where \(x_0\) and \(y_0\) are given specific values, this is our initial condition to find the exact curve that passes through the point \((x_0, y_0)\). This unique solution is tailored to the initial condition, making IVPs pivotal in simulations or models forecasting time dynamics, such as population growth, financial models, and physical phenomena predictions.
Understanding IVPs is critical as they ensure the solutions to differential equations are applicable and relevant in practical, real-world scenarios, providing precise models in fields ranging from biology to engineering.

Partial Derivatives

Partial derivatives form a fundamental part of multivariable calculus, extending the concept of derivatives to functions of several variables. Unlike a regular derivative which measures how a function changes as one single variable changes, a partial derivative focuses on one variable at a time, keeping the others constant.
In practice, if you have a function \(F(x, y)\), the partial derivative with respect to \(x\) is denoted as \(\frac{\partial F}{\partial x}\), and with respect to \(y\) as \(\frac{\partial F}{\partial y}\). These derivatives measure the rate of change of the function in the direction of each variable's axis. Such an approach helps unravel how changes in one dimension impact a multidimensional surface or curve.
Partial derivatives are central to identifying contour lines in geographical mapping or the gradient vector in optimization problems, highlighting the utility of this concept in both theoretical and applied contexts. In the context of exact differential equations, partial derivatives ensure we can find a function where applying these derivatives reconstructs our original equations \(M(x, y)\) and \(N(x, y)\), verifying its exactness with the given conditions.