Question

Consider the Euler equation, $$ x^{n} y^{(n)}+a_{n-1} x^{n-1} y^{(n-1)}+\cdots+a_{1} x y^{\prime}+a_{0} y=r(x) . $$ Substitute \(x=e^{t}\) and show that such a substitution reduces this to a DE with constant coefficients. In particular, solve \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x\).

Short answer

The substitution \(x=e^{t}\) reduces the Euler equation to a differential equation with constant coefficients, and the differential equation \( y^{\prime \prime}- 4 y^{\prime}+6 y = 1 \) can be solved to obtain \( C e^{2t} + D e^{-3t} + 1/6\), where C and D are constants.

Step-by-step solution

Substitution Step

First, replace \(x=e^{t}\). This implies that \( dx=e^{t} dt\) and \( d\^{n}x/dt^n = e^{t}*d^n e^{t}/dt^n\). Now substitute these in the Euler equation.

Simplification Step

After substitution, we get \[ e^{nt} y^{(n)}+a_{n-1} e^{(n-1)t} y^{(n-1)}+\cdots+a_{1} e^{t} y^{\prime}+a_{0} y=e^{t}r(e^{t}) \]Divide both sides by \(e^{nt}\), this simplifies the equation to an equation with constant coefficients.

Application to Given Euler Equation

Using substitution on the specific Euler equation: \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x\), we get \[ e^{2t} y^{\prime \prime}-4 e^{t} y^{\prime}+6 y=e^{t}.\] After dividing by \(e^{2t}\), we have a simpler differential equation, which is \( y^{\prime \prime}- 4 y^{\prime}+6 y = 1\).

Solution Step

This new differential equation is now easier to solve. The solution is \( y = Ce^{2t} + De^{-3t} + 1/6 \), where C and D are constants.

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. In essence, it describes how a certain quantity changes over time or space. Differential equations are fundamental in various areas such as physics, engineering, economics, and biology because they model the behavior of dynamic systems. For example, they can represent the growth rate of a population, the movement of a charged particle, or the change in velocity of a moving car. The complexity of differential equations can vary greatly; some can be solved analytically with pen and paper, while others require sophisticated numerical methods and computational power.

When it comes to solving differential equations, there are several approaches, but they all require understanding the equation's structure, analyzing the types of derivatives involved, and applying the appropriate solution method. The Euler equation provided in the original exercise is a particular type of differential equation commonly encountered in the study of advanced mathematics and physics.

Constant Coefficients

The term constant coefficients in the context of differential equations refers to equations where the coefficients of the derivatives are constants—that is, they do not change in respect to the independent variable. Such equations are especially notable because they are easier to solve compared to those with variable coefficients. For instance, an equation like \( y'' + p y' + q y = 0 \), where \( p \) and \( q \) are constants, can be systematically solved using characteristic equations or other standard methods such as undetermined coefficients or variation of parameters.

Differential equations with constant coefficients have clear-cut solution strategies, which often involve exponential functions or trigonometric functions. Understanding how to deal with constant coefficients is crucial for students because many physical phenomena lead to equations of this type, allowing for quick and precise solutions.

Substitution Method

The substitution method is a powerful technique for simplifying and solving differential equations. It involves replacing the original independent variable with a new one, thereby potentially transforming a complex equation into a simpler form. In the exercise at hand, substituting \( x = e^t \) leads to a change of variables which helps to convert the Euler equation, which has variable coefficients, into one with constant coefficients.

Substitution can be particularly helpful when dealing with differential equations possessing certain symmetries or characteristics that become more obvious after the transformation. It's worth noting that the choice of substitution is not arbitrary and requires some insight into the nature of the equation. In some cases, like the Euler equation, standard substitutions are well-known, whereas, in other scenarios, finding the right substitution could be part of the problem-solving challenge.

Differential Equation Solutions

The solutions to differential equations can take many forms, including explicit functions, parametric equations, or series. The goal of solving a differential equation is to find the relationship between the dependent and independent variables that satisfies the given equation. Solutions are not always unique; there may be a family of solutions depending on the initial or boundary conditions.

In the context of the Euler equation example provided, the substitution method leads us to a second-order linear differential equation with constant coefficients, which has a standard solution method. These solutions typically include homogeneous solutions, determined by the characteristic equation, and particular solutions, which account for any nonhomogeneity, such as the presence of the term \( e^t \) on the right side. The final expression \( y = Ce^{2t} + De^{-3t} + \frac{1}{6} \) exemplifies a full solution—consisting of a complementary function \( Ce^{2t} + De^{-3t} \), which is the general solution to the homogeneous equation, and a particular solution \( \frac{1}{6} \), which corresponds to the nonhomogeneous part of the equation.