Problem 57 - Differential Equations and Linear Algebra

Question

Show that the change of variables \(V=x y\) transforms the differential equation $$ \frac{d y}{d x}=\frac{y}{x} F(x y) $$ into the separable differential equation $$ \frac{1}{V[F(V)+1]} \frac{d V}{d x}=\frac{1}{x} $$

Step-by-Step Solution

Answer

Using the change of variables \(V = xy\), the given differential equation can be transformed into a separable differential equation as follows: 1. Rewrite the given differential equation in terms of \(V\): \(\frac{d(\frac{V}{x})}{dx} = \frac{\frac{V}{x}}{x}F(\frac{V}{x}\cdot x)\) 2. Find the derivative of \(\frac{V}{x}\) with respect to \(x\): \(\frac{d(\frac{V}{x})}{dx} = \frac{1}{x}\frac{dV}{dx}\) 3. Plug the derivative into the differential equation: \(\frac{1}{x}\frac{dV}{dx} = \frac{\frac{V}{x}}{x}F(V)\) 4. Simplify the resulting equation: \(\frac{1}{V[F(V)+1]}\frac{dV}{dx} = \frac{1}{x}\) Hence, the separable differential equation is \(\frac{1}{V[F(V)+1]}\frac{dV}{dx} = \frac{1}{x}\).

Step 1: Rewrite the given differential equation in terms of V

Let us begin by rewriting the given differential equation, replacing \(y\) with \(\frac{V}{x}\): \[ \frac{d\left(\frac{V}{x}\right)}{dx} = \frac{\frac{V}{x}}{x} F\left(x\frac{V}{x}\right) \] In the next step, we will find the derivative with respect to \(x\) of both sides of the equation.

Step 2: Find the derivative of \(\frac{V}{x}\) with respect to \(x\)

Our next goal is to find the derivative of \(\frac{V}{x}\) with respect to \(x\). Since \(V\) is a function of \(x\) (through the variable \(y\)), we will have to use the chain rule: \[ \frac{d\left(\frac{V}{x}\right)}{dx} = \frac{dV}{dx}\cdot\frac{d\left(\frac{V}{x}\right)}{dV} \] Differentiating \(\frac{V}{x}\) with respect to \(V\), we get: \[ \frac{d\left(\frac{V}{x}\right)}{dV} = \frac{1}{x} \] Now, substituting this back into the expression above, we obtain: \[ \frac{d\left(\frac{V}{x}\right)}{dx} = \frac{1}{x}\frac{dV}{dx} \]

Step 3: Plug the derivative into the differential equation

Now, we can substitute the expression we found for the derivative, \(\frac{1}{x}\frac{dV}{dx}\), into the differential equation from Step 1: \[ \frac{1}{x}\frac{dV}{dx} = \frac{\frac{V}{x}}{x} F\left(\frac{V}{x}\cdot x\right) \]

Step 4: Simplify the resulting equation

Observe that the argument of the function \(F\) now simplifies to simply \(V\): \[ \frac{1}{x}\frac{dV}{dx} = \frac{\frac{V}{x}}{x} F\left(V\right) \] Now, we can separate variables \(x\) and \(V\) by multiplying both sides by \(x^2\) and dividing both sides by \(F(V)+1\): \[ \frac{1}{V[F(V)+1]}\frac{dV}{dx} = \frac{1}{x} \] The differential equation is now successfully transformed into a separable differential equation.

Key Concepts

Separable Differential Equations

Separable differential equations are a special class of differential equations where the variables can be separated on opposite sides of the equation. This allows for straightforward integration, making them easier to solve.
A typical form of a separable differential equation is \( \frac{dy}{dx} = g(y) h(x) \), where the functions of \( y \) and \( x \) can be isolated.
To solve a separable differential equation, follow these steps:

• Move all terms involving \( y \) to one side of the equation, and all terms involving \( x \) to the other side.
• Integrate both sides with respect to their variables.
• Solve for the function \( y \), if needed, to find the particular or general solution.

By transforming the original equation in our problem using the change of variables technique, we ended up with a separable differential equation. This step is crucial as it allows us to utilize these simple methods of integration to solve the equation effectively.

Transformation of Differential Equations

The transformation of differential equations involves rewriting the original equation into a new form that may be simpler to solve.
This is often done by introducing a change of variables, as seen in the problem where \( V = xy \) was used.
Such transformations can make difficult equations tractable by simplifying complex relationships between variables.

• Choosing an appropriate substitution can dramatically change the form of the differential equation.
• Once transformed, the new equation may be of a type (like separable) that is easier to handle.
• Transformations often rely on recognizing patterns or symmetries in the equation.

Through transformation, a complex non-separable equation becomes a simpler separable one, allowing us to solve it using integration techniques. This approach is particularly useful when tackling equations that are not easily solvable in their original form.

Chain Rule in Calculus

The chain rule is a fundamental concept in calculus, especially useful when dealing with composite functions.
It is used to differentiate a function that depends on another function, and it plays a critical role in finding derivatives in various situations, including our problem solution.
For a function \( y = f(g(x)) \), the chain rule is expressed as:

• \( \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} \)
• This means you first differentiate \( f \) with respect to \( g \), and then multiply by the derivative of \( g \) with respect to \( x \).
• The chain rule helps in breaking down complex derivatives into simpler parts.

In our exercise, using the chain rule allowed us to compute the derivative of \( \frac{V}{x} \) with respect to \( x \), facilitating the transformation process of the differential equation. This technique is instrumental in many calculus problems, helping to untangle derivatives in nested functions.