Question

In Exercises \(1-17\) determine which equations are exact and solve them. $$ \left(-2 y^{2} \sin x+3 y^{3}-2 x\right) d x+\left(4 y \cos x+9 x y^{2}\right) d y=0 $$

Short answer

Answer: No, the given differential equation is not exact.

Step-by-step solution

Identify the functions

The given equation is: $$ \left(-2 y^{2} \sin x+3 y^{3}-2 x\right) d x+\left(4 y \cos x+9 x y^{2}\right) d y=0 $$ Now, identify the functions: $$ M(x, y) = -2y^2\sin x + 3y^3 - 2x \\ N(x, y) = 4y\cos x + 9xy^2 $$ and their partial derivatives. $$ \frac{\partial M}{\partial y} = -4y\sin x + 9y^2 \\ \frac{\partial N}{\partial x} = 4y\cos x + 9y^2 $$

Check if the equation is exact

An equation is exact when: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$ In our case: $$ -4y\sin x + 9y^2 = 4y\cos x + 9y^2 $$ As we can see, the equation is not exact. Since the equation is not exact, we cannot directly find a total differential of a function. To find a solution, we must use other techniques such as integrating factors, variable separation, or numerical methods. In this case, since the equation is not exact, we are unable to solve it by creating a step-by-step solution as requested.

Key concepts

Partial Derivatives

Partial derivatives are fundamental concepts in calculus, especially when dealing with functions of several variables. They represent the rate of change of a function with respect to one of its variables, while treating all other variables as constants. In the context of an exact differential equation, we identify functions like \( M(x, y) \) and \( N(x, y) \). These functions are part of the equation's structure:

• \( M(x, y) \) is associated with \( dx \)
• \( N(x, y) \) is associated with \( dy \)

To determine if a differential equation is exact, we calculate the partial derivatives:

• \( \frac{\partial M}{\partial y} \) for \( M(x, y) \)
• \( \frac{\partial N}{\partial x} \) for \( N(x, y) \)

If these derivatives are equal, then the equation is exact, meaning it can be transformed into a total derivative of some function. If they are not equal, as shown in the exercise, the equation is not exact. This indicates that the solution requires further techniques.

Integrating Factors

When an equation is not exact, integrating factors are often employed to make it exact. An integrating factor is a function that, when multiplied with the original terms of the differential equation, alters it into an exact equation. This leads to finding a potential function whose differential corresponds to the given equation.

Finding the right integrating factor isn't straightforward and depends on the form of the equation. It involves inspecting the terms carefully and sometimes relying on assumptions or known transformations:

• For some cases, the relationship or dependency on one of the variables can give a hint.
• One can also use standard forms or known expressions to simplify the process.

The goal is to transform the partial derivatives of the altered equation so that they satisfy the condition for exactness: \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). Once the equation is exact, it becomes solvable by methods of finding a function whose total differential covers both terms.

Numerical Methods

Numerical methods provide a powerful alternative when analytical methods like exactness or integrating factors are challenging to apply. These methods are especially useful for equations that cannot be solved conveniently by hand or when an approximate solution suffices.

Common numerical approaches include:

• Euler's method
• Runge-Kutta methods
• Finite difference methods

These techniques allow computers to approximate solutions to differential equations. While they may not provide the exact function, they offer calculated values at specific points along the solution curve. Numerical methods are invaluable in engineering, physics, and applied sciences where precise analytic solutions are difficult due to the complexity of the equations.

Variable Separation

Variable Separation, often referred to as "separation of variables," is a method for solving differential equations by expressing them in a form where each variable can be rearranged to one side of the equation. This is primarily applicable to simpler equations involving multiplication, where separating allows integration on both sides.

The steps are as follows:

• Rearrange the equation such that each variable and its differentials are on opposite sides.
• Integrate both sides with respect to their respective variables.

This results in two integrals that correspond to each variable. Although separation of variables is not directly applicable to the provided exercise due to its complexity, understanding this method is essential as it forms the basis for analyzing and simplifying differential equations in more straightforward scenarios.