Question

Find the general solution of the given Euler equation on \((0, \infty)\). $$ x^{2} y^{\prime \prime}+x y^{\prime}+y=0 $$

Short answer

#Short Answer# To solve the given Euler equation, first a substitution of variables is made to transform it into a second-order constant coefficient linear differential equation. The general solution for the transformed equation is found and then the reverse substitution is applied to find the final general solution for the Euler equation. The steps are: 1. Change the variables to transform the equation. 2. Substitute the new variables and simplify the equation. 3. Reduce the transformed equation to a second-order constant coefficient linear differential equation. 4. Solve the second-order constant coefficient linear differential equation and make a reverse substitution to find the general solution for the original Euler equation.

Step-by-step solution

- Change the variables

To change the variables, we can define a new function and a new independent variable as: $$ y(x) = x^{m} u(t) $$ $$ t = \ln x $$ Taking derivatives of the new function \(y(x)\): $$ y'(x) = m x^{m-1} u(t) + x^{m} u'(t) $$ $$ y''(x) = m(m - 1) x^{m - 2} u(t) + 2 m x^{m-1} u'(t) + x^m u''(t) $$ Now, replace \(y(x)\), \(y'(x)\), and \(y''(x)\) with the new function and its derivatives in the given Euler equation.

- Substitute and Simplify

Replace and simplify the given Euler equation: $$ x^2 [m(m - 1) x^{m - 2} u(t) + 2 m x^{m-1} u'(t) + x^m u''(t)] + x [m x^{m-1} u(t) + x^{m} u'(t)] + x^m u(t) = 0 $$ Multiply and simplify terms: $$ m(m - 1) x^{m} u(t) + 2 m x^{m} u'(t) + x^{m+1} u''(t) + m x^{m} u(t) + x^{m+1} u'(t) + x^m u(t) = 0 $$ Divide every term by \(x^m\): $$ m(m - 1) u(t) + 2 m u'(t) + x u''(t) + m u(t) + x u'(t) + u(t) = 0 $$ Now, substitute \(x = e^t\) and simplify: $$ m(m - 1) u(t) + 2 m u'(t) + e^t u''(t) + m u(t) + e^t u'(t) + u(t) = 0 $$

- Reduce to Second-Order Constant Coefficient Linear Differential Equation

Now, the transformed equation is: $$ e^t u''(t) + (2m + e^t)u'(t) + (m^2 + m + 1) u(t) = 0 $$ The general form of the second-order constant coefficient linear differential equation is: $$ y'' + Ay' + By = 0 $$ Comparing our transformed equation with the general form, we can see that: $$ A = 2m + e^t, \quad B = m^2 + m + 1 $$

- Solve the Second-Order Constant Coefficient Linear Differential Equation

This is where the limit of step-by-step general solution reaches. In this case, we'd need to investigate the form of \(u(t)\) based upon the possible types of solutions (real distinct roots, repeated roots, or complex conjugate roots) combined with the structure of the coefficients in this transformed equation. However, once you find the general solution for \(u(t)\), reverse the variable substitution to find the general solution for the original Euler equation using the initial substitution: $$ y(x) = x^{m} u(\ln x) $$

Key concepts

Understanding Differential Equations

Differential equations form the backbone of a range of disciplines including engineering, physics, and economics. They describe relationships involving rates of change and are pivotal in modeling real-world systems. An equation that relates a function with its derivatives is called a differential equation.

When confronted with a differential equation like the Euler equation given in the exercise, the objective is to find the function that satisfies the equation, typically referred to as the 'solution' of the equation. In this context, the solution \(y(x)\) is not a single value but rather a function that will satisfy the equation for all \(x\) within a specified domain, in this case \(x > 0\).

Finding the general solution of a differential equation can often involve sophisticated mathematical techniques. The process requires identifying the type of differential equation at hand—whether it is ordinary or partial, linear or nonlinear—and then applying appropriate methods and strategies, such as variable substitution, to solve it.

Variable Substitution

Variable substitution is a technique used to simplify differential equations, making them easier to solve. It's akin to choosing a more convenient coordinate system in which the problem looks simpler. In the case of the given Euler equation, a specific type of substitution is used, which involves both the function \(y(x)\) and the independent variable \(x\).

The substitution provided in the exercise links \(x\) and a new variable \(t\) with the natural logarithm. This is a common trick for Euler equations since it turns them into differential equations with constant coefficients after the subsequent substitutions and simplifications. Moreover, the exercise involves expressing \(y\) as \(x^{m} u(t)\), implying that \(y\)’s behavior is tied to the behavior of \(u(t)\) and the power of \(x\).

The rationale behind this substitution is based on properties of logarithms and exponents that simplify the expression and its derivatives. Once the substitutions are made, the original equation is transformed into one that is more straightforward to solve.

Tackling Second-Order Constant Coefficient Linear Differential Equations

Second-order constant coefficient linear differential equations are a specific form of differential equations characterized by the presence of a second derivative and constant terms multiplying the function and its first derivative. The general form is \(y'' + Ay' + By = 0\), where \(A\) and \(B\) are constants.

This type of equation is significant because it has well-established solution methods and directly applies to a variety of physical systems, like harmonic oscillators in physics. The solution approach often hinges on finding the roots of the characteristic equation, which is a quadratic equation derived from the coefficients of \(y\), \(y'\), and \(y''\). Depending on the nature of these roots—real or complex—the form of the general solution varies.

In the exercise, the transformation of the Euler equation via substitution reduces it to this form with non-constant coefficients \(A\) and \(B\), which are functions of the new variable \(t\). Despite this complexity, the equation can be approached with strategies similar to those used for constant coefficients by focusing on the transformed function \(u(t)\). Ultimately, once the solution for \(u(t)\) is found and the scenario of its roots is understood, it can be re-substituted to find the original function \(y(x)\).