Solving the Differential Equation x^2y'' + 10xy' + 8y = 0 Using Cauchy Euler's Rule

In this article, we will explore how to solve the second-order linear differential equation x^2y'' + 10xy' + 8y = 0 using the Cauchy Euler's rule. This method allows us to find the complementary solution (y_c) and the particular solution (y_p) of the equation.

Step 1: Identify the Equation Type

The given differential equation is a second-order linear differential equation with constant coefficients. The general form of such an equation is:

ax^2y'' + bxy' + cy = f(x)

In our case, the coefficients are:

• a = 1
• b = 10
• c = 8
• f(x) = 0 (the equation is homogeneous)

Step 2: Apply Cauchy Euler's Rule

Cauchy Euler's rule is a method used to solve homogeneous second-order linear differential equations with constant coefficients. The steps are as follows:

1. Characteristic Equation: Construct the characteristic equation by substituting y = x^r into the differential equation and simplifying: x^2(r^2 - r) + 10x(r) + 8 = 0 Dividing both sides by x^2, we get: r^2 - r + 10r + 8 = 0 Rearranging, we have: r^2 + 9r + 8 = 0

2. Roots of the Characteristic Equation: Solve the characteristic equation to find the roots, r_1 and r_2: r_1 = -1 r_2 = -8

3. Complementary Solution (y_c): The complementary solution is given by: y_c = C_1 x^{r_1} + C_2 x^{r_2} Substituting the values of r_1 and r_2, we get: y_c = C_1 x^{-1} + C_2 x^{-8}

4. Particular Solution (y_p): Since the given equation is homogeneous (i.e., f(x) = 0), the particular solution is y_p = 0.

Therefore, the general solution to the differential equation x^2y'' + 10xy' + 8y = 0 is: y = y_c + y_p = C_1 x^{-1} + C_2 x^{-8}

The constants C_1 and C_2 can be determined based on the given initial conditions or boundary conditions.

I hope this article has helped you understand how to solve the given differential equation using Cauchy Euler's rule. If you have any further questions or need additional assistance, feel free to ask.