Question

Solve the differential equation \(y y^{\prime 2}+2 x y^{\prime}-y=0\) by changing from variables \(y, x\), to \(r, x\), where \(y^{2}=r^{2}-x^{2}\); then \(y y^{\prime}=r^{\prime}-x\).

Short answer

Apply the change of variables and solve the simplified equation.

Step-by-step solution

Change of Variables

Convert the given function from variables \(y, x\) to \(r, x\) using the given relationship \(y^2 = r^2 - x^2\).

Express y in Terms of r and x

From \(y^2 = r^2 - x^2\), solve for \(y\), resulting in \(y = \sqrt{r^2 - x^2}\).

Calculate the Derivative

Determine \(y'\) by differentiating \(y = \sqrt{r^2 - x^2}\) with respect to \(x\). Use the chain rule to get \(y' = \frac{-x}{\sqrt{r^2 - x^2}}\).

Express \(yy'\) in Terms of r and its Derivatives

Given \(yy' = r' - x\), substitute \(y'\) to express \(yy'\) in the appropriate terms. This gives \(y \cdot \frac{-x}{\sqrt{r^2 - x^2}} = R' - x\).

Simplify the Original Equation

Substitute all the expressions derived above into the original differential equation \(y y^{\frac{2}} +2x y' - y = 0\). Simplify and solve for \(r\).

Solve the Differential Equation

Given the simplifications and changes of variables, solve the resulting equation for \(r\) and subsequently for \(y\).

Key concepts

Change of Variables

Changing variables is a powerful technique in solving differential equations where direct methods seem cumbersome. Here, the given differential equation involves variables \(y, x\). By introducing a new variable \(r\), we can potentially simplify the equation. We use the relationship \(y^2 = r^2 - x^2\). This substitution changes the functions and potentially reduces the complexity. This transformation helps in re-framing the problem into a more manageable form.
Consider how the relationship \(y^2 = r^2 - x^2\) mandates that each side be equal for all \(x\) and thus encapsulates the essence of the transformation.

Chain Rule

The chain rule is essential in calculus for finding derivatives of composite functions. In this context, it's crucial for differentiating complex expressions indirectly.
Here, we need to find the derivative of \(y = \sqrt{r^2 - x^2}\). Using the chain rule, we consider \(r\) as a function of \(x\). Consequently, the derivative of \(y\) with respect to \(x\) becomes \(-x / \sqrt{r^2 - x^2}\).
This step is significant as it translates the derivatives from one set of variables to another, paving the way for further simplification.

Differentiation

Differentiation involves finding the rate at which a function changes. In the given problem, differentiating \(y\) with respect to \(x\) is crucial to progress further.
The direct differentiation of \(y = \sqrt{r^2 - x^2}\) gives us \(y' = \frac{-x}{\sqrt{r^2 - x^2}}\). This step is directly influenced by the chain rule. Here, the negative sign indicates the directional change of the slope concerning \(x\).
This computation aligns with transforming the original differential equation into a new form that is easier to solve.

Simplification

Simplification is the process of rewriting mathematical expressions in a more readable or solvable form. Once we have our derivatives, we substitute them back into the original equation \(y y^{\frac{2}} + 2x y' - y = 0\).
This requires careful algebraic manipulation to minimize complexity. In this scenario, expressing \(yy' = r' - x\) and combining terms accordingly simplifies the differential equation.
Typically, simplification reduces terms to expose easier paths to solutions, making the final integration step more straightforward.

Integration

Integration is the reverse process of differentiation, and it’s often used to solve differential equations. In this transformed problem, the final goal is to integrate the simplified equation for \(r\) to find its behaviour over \(x\).
Based on the substitution and derived expressions, integrating helps us revert to the original variable. In our case, solving for \(r\) and substituting back to find \(y\) completes the solution process.
Integration aggregates the changes of a function over an interval, providing a comprehensive solution.