Question

Suppose that \(x^{r}_{1}\) and \(x^{r_{2}}\) are solutions of an Euler equation for \(x>0,\) where \(r_{1} \neq r_{2},\) and \(r_{1}\) is an integer. According to Eq. ( 24) the general solution in any interval not containing the origin is \(y=c_{1}|x|^{r_{1}}+c_{2}|x|^{r_{2}} .\) Show that the general solution can also be written as \(y=k_{1} x^{r}_{1}+k_{2}|x|^{r_{2}} .\) Hint: Show by a proper choice of constants that the expressions are identical for \(x>0,\) and by a different choice of constants that they are identical for \(x<0 .\)

Short answer

Question: Show that the general solution of an Euler equation can be written as \(y = k_1 x^{r_1} + k_2 |x|^{r_2}\) for the given conditions. Answer: Choose constants \(k_1 = c_1(-1)^{r_1}\) and \(k_2 = c_2\), then the general solution can be written as \(y = k_1 x^{r_1} + k_2 |x|^{r_2}\) which becomes identical with \(y = c_1 |x|^{r_1} + c_2 |x|^{r_2}\) for both positive and negative \(x\) values.

Step-by-step solution

Introduce the given equations

We have the general solution of an Euler equation as: $$ y = c_1 |x|^{r_1} + c_2 |x|^{r_2}. $$ We need to show that it can also be written as: $$ y = k_1 x^{r_1} + k_2 |x|^{r_2}. $$

Determine the expression for x > 0

When \(x > 0\), both the expressions can be rewritten as: 1. \(y = c_1 x^{r_1} + c_2 x^{r_2}\), and 2. \(y = k_1 x^{r_1} + k_2 x^{r_2}\) (since \(|x|=x\) when \(x>0\)). Now we can choose the constants \(k_1\) and \(k_2\) such that $$ k_1 = c_1 \quad \text{and} \quad k_2 = c_2. $$ Hence, both expressions become identical for \(x > 0\).

Determine the expression for x < 0

When \(x < 0\), the expressions become: 1. \(y = c_1 (-x)^{r_1} + c_2 (-x)^{r_2}\), and 2. \(y = k_1 (-x)^{r_1} + k_2 |x|^{r_2}\) (since \(|x|= -x\) when \(x<0\)). Now, let's choose the constants \(k_1\) and \(k_2\) such that $$ k_1 = c_1(-1)^{r_1} \quad \text{and} \quad k_2 = c_2. $$ Thus, the expressions become identical for \(x < 0\).

Conclude by showing the equivalence of the expression for all x values

We have demonstrated that by choosing appropriate constants \(k_1\) and \(k_2\), the general solutions can be written in the given form and are identical for both positive and negative \(x\) values. Therefore, we have shown that the general solution can be written as: $$ y = k_1 x^{r_1} + k_2 |x|^{r_2}. $$

Key concepts

Euler Equation

An Euler Equation is a type of differential equation that has special characteristics. It often appears in the form \( ax^2y'' + bxy' + cy = 0 \), where the powers of x match the order of differentiation. These equations are uniquely suited to transformations that simplify them to constant coefficient differential equations, making solving them easier.

Understanding Euler equations is important because they help model various scientific and engineering phenomenons. They usually involve exponential functions due to their structural characteristics. This quality makes Euler equations efficient in demonstrating the growth or decay processes in natural and engineering scenarios.

General Solution

The general solution to an Euler equation provides a formula to find a family of functions that satisfy the differential equation. For an Euler equation, the solution is expressed in terms of constants and powers of x. For example, given as \( y = c_1 |x|^{r_1} + c_2 |x|^{r_2} \), it represents a combination of solutions with undetermined constants \( c_1 \) and \( c_2 \).

The solution hints at the presence of two distinct roots, \( r_1 \) and \( r_2 \), showing that different behaviors depending on the system conditions can be captured. Ensuring a good grasp of constructing solutions with these forms allows tackling more complex systems effectively by providing a structural solution approach.

Constants Selection

Selecting the correct constants in the general solution is crucial for the equation's applicability in modeling a scenario accurately. In step two of the original solution, it was seen that altering the constants \( k_1 \) and \( k_2 \) to match the existing constants \( c_1 \) and \( c_2 \) allowed both expressions to equal each other for positive x values.

This highlights the importance of constants in maintaining the same functional forms of equations across different conditions. Matching constants correctly are fundamental to ensuring that the solution accurately reflects the physical constraints or initial conditions given in an actual scenario. It allows theoretical models to align closely with real-world situations.

Negative and Positive Values Handling

Handling negative and positive values in Euler equations requires careful manipulation of absolute values to ensure the correctness of the solution. As seen in steps two and three, the absolute value affects the treatment of the power terms depending on whether x is positive or negative.

For \( x > 0 \), the simplification \( |x| = x \) directly makes the expressions straightforward, resembling traditional polynomial forms. Conversely, for \( x < 0 \), \( |x| = -x \) necessitates adjustments with constants and powers, maintaining a valid solution that mirrors reality.

Understanding this distinction in how x is handled ensures you adapt the general solution to fit both positive and negative domains, signifying how mathematical models adjust to encompass broader conditions. This adaptability is contingent on knowing how to appropriately evaluate and manipulate the signs in your computations.