Problem 21 - Differential Equations and Linear Algebra

Question

Solve the given differential equation on the interval \(x>0 .\) [Remember to put the equation in standard form.] \(x^{2} y^{\prime \prime}-(2 m-1) x y^{\prime}+m^{2} y=x^{m}(\ln x)^{k},\) where \(m, k\) are constants.

Step-by-Step Solution

Answer

The short version of the answer to the given differential equation is to first obtain the complementary function, \(y_c(x) = C_1 x^{r_1} + C_2 x^{r_2}\), by solving the homogeneous equation. Then find the particular integral, \(y_p(x)\), using the method of variation of parameters. Finally, combine the complementary function and the particular integral to find the general solution that satisfies the given differential equation on the interval \(x > 0\), which has the form \(y(x) = y_c(x) + y_p(x)\).

Step 1: Determine the homogeneous equation

To find the complementary function, first set the right-hand side of the given equation to 0: \(x^2 y'' - (2m - 1) xy' + m^2 y = 0\) This is now a homogeneous second-order linear differential equation in which we can look for the complementary function, yc(x), that satisfies the homogeneous part of the given equation. Step 2: Solve the homogeneous equation

Step 2: Assume a power-law solution

To solve the homogeneous equation, assume a power-law solution of the form: \(y_{c}(x) = x^r\) with y'_c(x) = r * x^(r-1) and y''_c(x) = r(r-1) * x^(r-2). Plug these expressions for y_c(x), y'_c(x), and y''_c(x) into the homogeneous equation and simplify. Step 3: Obtain the complementary function

Step 3: Find the r values

After substituting and simplifying the homogeneous equation, we'll find the values of r that satisfy the equation. This will give us the complementary function in the form: \(y_c(x) = C_1 x^{r_1} + C_2 x^{r_2}\) where C1 and C2 are arbitrary constants. Step 4: Applying variation of parameters

Step 4: Find the particular integral

Now we need to find the particular integral, yp(x), that incorporates the given inhomogeneous term \(x^m (\ln x)^k\). To do this, we will use the method of variation of parameters. Apply this method to the given differential equation, obtaining an equation for yp(x). Step 5: Obtain the general solution

Step 5: Combine complementary function and particular integral

The general solution of the given inhomogeneous differential equation is the sum of the complementary function and the particular integral: \(y(x) = y_c(x) + y_p(x)\) Once we found both yc(x) and yp(x), we can write the general solution for the differential equation. This will be the unique solution that satisfies the given differential equation on the interval \(x > 0\).

Key Concepts

Homogeneous Equation

In the study of differential equations, a homogeneous equation is one where all terms are a function of the unknown variable and its derivatives, with no free-standing constant terms. The equation in the solved problem, \(x^2 y'' - (2m - 1) xy' + m^2 y = 0\), is an example of a homogeneous linear differential equation. It is crucial to understand that the 'homogeneous' part refers to the property that if you have a solution, any constant multiple of it is also a solution. For differential equations like this, it means the solution is composed solely of terms that are derivative dependent, molded around a base form that fits a standard solution pattern. Such equations often qualify for standard solving techniques, such as assuming solutions with specific forms, which can lead to easily finding the complementary solution for the equation.

Power-law Solution

One common technique for solving second-order linear homogeneous differential equations is assuming a solution that follows a power law. In simpler terms, you guess that the solution looks like a power of \(x\). For the problem at hand, the assumed power-law solution is of the form \(y_c(x) = x^r\).This technique simplifies the equation drastically. When you substitute these terms back into the differential equation, you often derive a simple algebraic equation to find \(r\), the power. By solving this, you can determine the values of \(r\) that satisfy the differential equation. In our example, the values of \(r\) will help create the complementary function: \(y_c(x) = C_1 x^{r_1} + C_2 x^{r_2}\), where \(C_1\) and \(C_2\) are constants determined by boundary conditions or initial conditions.

Variation of Parameters

Variation of parameters is a sophisticated yet efficient method for determining a particular solution to a non-homogeneous differential equation. This method involves using the previously solved complementary solution and adjusting it to incorporate the non-homogeneous part of the equation. After finding the complementary function, \(y_c(x)\), we need a particular solution, \(y_p(x)\), which is tailored to accommodate the non-homogeneous part, \(x^m (\ln x)^k\), in this example. It does so by generalizing the constants in the complementary function into functions of \(x\) and solving a corresponding system of equations. Eventually, this method gives you a particular solution that, when added to the complementary solution, forms the general solution of the differential equation: \(y(x) = y_c(x) + y_p(x)\). This holistic solution captures both how the system naturally behaves (homogeneously) and how it responds to external forces or inputs (non-homogeneously).