Chapter 8: Problem 24

Identify each of the differential equations as to type (for example, separable, linear first order, linear second order, etc.), and then solve it. \(x\left(y y^{\prime \prime}+y^{\prime 2}\right)=y y^{\prime}\) Hint: Iet \(u=1 x\).

Short Answer

Expert verified

Nonlinear second-order differential equation, solve using substitution and integration for transformation of simpler equation

Step by step solution

Identify the type of differential equation

Given the differential equation: \[ x\left(yy^{\prime \prime} + y^{\prime 2}\right) = yy^{\prime} \] This is a nonlinear second-order differential equation due to the presence of the term \(yy^{\prime \prime}\) and the product of derivatives \(y^{\prime 2}\).

Apply the substitution

Use the hint provided, where \( u = y/x \). This implies that: \[ y = ux \quad and \quad y^{\prime} = u + xu^{\prime} \] Substituting these into the original equation to simplify.

Simplify the equation

Substitute \( y = ux \) and \( y^{\prime} = u + xu^{\prime} \) into the original equation: \[ x\left(y(y^{\prime \prime}) + y^{\prime 2} \right) = yy^{\prime} \] This becomes: \[ x\left((ux)(xu^{\text{\textbrokenbar}} + u) + (u + xu^{\text{\textbrokenbar}})^2\right) = (ux)(u + xu^{\text{\textbrokenbar}}) \]

Transform and solve the simplified equation

Further simplify and separate variables to solve for \( u \). First simplify the expressions within the equation: \[ x\left(uxxu^{\text{\textbrokenbar}} + uxu + u^2 + 2xu^2u^{\text{\textbrokenbar}} + (xu^{\text{\textbrokenbar}})^2\right)= xuu + x^2 u^{\text{\textbrokenbar}}) \]Combine like terms to manage the transformed differential equation.

Integrate to find the solution for the transformed differential equation

Solve the obtained differential by integrating and reversing the substitution \(u = y/x\). One receives: \[y/x =\int 1/dx \] leading \[ y = x^\textplainform \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nonlinear Second-Order Differential Equations

In the given problem, we are dealing with a nonlinear second-order differential equation. These types of equations have derivatives that are not linear, and the highest derivative is of the second order. In our exercise, this is clearly seen in the equation: \[ x(yy^{\text{\textbrokenbar}} + y^{\text{\textbrokenbar} 2}) = yy^{\text{\textbrokenbar}}. \] Nonlinearity is indicated by the product of derivatives and the presence of terms like \( yy^{\text{\textbrokenbar}} \). Solving such equations often requires specific techniques and sometimes creative substitutions.

Substitution Method

Substitution is a powerful technique to simplify complex differential equations. Here, the hint suggests using \( u = y/x \). This transformation makes the equation more manageable. Rewriting \( y \) and its derivatives in terms of \( u \) gives: \[ y = ux \quad \text{and} \quad y^{\text{\textbrokenbar}} = u + xu^{\text{\textbrokenbar}}. \] By substituting these into the original equation, it changes the form and reduces its complexity. After substitution and simplifying, the new form of the equation often becomes more approachable.

Integration

Integration is a core step in solving differential equations, especially after simplifying through substitution. Here, once the equation is simplified, it is separated to allow integration: \[ \frac{dy}{dx} = f(u). \] Integrating with respect to \(x\) helps find solutions in terms of \(u\), and then reversing the substitution \( u = y/x \) reverts the solution back to the original variables. Integration bridges the gap between transforming the differential equation and getting back to the function of interest. It’s essential to perform each integration step carefully to ensure the solution aligns with the initial conditions and requirements of the problem.