The given differential equation is:

$x^2 y'' + 7x y' + 9y = (22 + 9) \ln x$

This type of differential equation is suitable for the Euler-Cauchy (or Cauchy-Euler) method, which is designed to solve linear second-order differential equations of the form:

$x^2 y'' + ax y' + b y = g(x)$

Steps to Solve Using the Euler-Cauchy Method

1. Identify the coefficients of $y'', y',$ and $y$. Here:

   • Coefficient of $y''$: $x^2$
   • Coefficient of $y'$: $7x$
   • Coefficient of $y$: $9$

   Therefore, the equation can be written in standard form with $a = 7$ and $b = 9$.

2. Solve the Associated Homogeneous Equation: The associated homogeneous equation is: $x^2 y'' + 7x y' + 9y = 0$

   Assume a solution of the form $y = x^r$, and substitute into the homogeneous equation to get the characteristic equation: $r(r - 1) + 7r + 9 = 0$

   Simplify this characteristic equation and solve for $r$ to find the values that satisfy the homogeneous solution.

3. Find the Particular Solution:
   For the non-homogeneous part $(22 + 9) \ln x = 31 \ln x$, we’ll use the method of undetermined coefficients or variation of parameters to find a particular solution that incorporates $\ln x$.

Would you like the detailed step-by-step solution through each part, including solving the characteristic equation and finding the particular solution?

Related Questions

1. How do you solve the characteristic equation for an Euler-Cauchy differential equation?
2. What is the method of undetermined coefficients, and how is it applied to this equation?
3. How does the solution change if the non-homogeneous term is a polynomial instead of $\ln x$?
4. Can this type of differential equation have complex roots, and how would that affect the solution?
5. What are the applications of Euler-Cauchy equations in physics and engineering?

Tip

For Euler-Cauchy equations, always check if the non-homogeneous term suggests a natural choice for the form of the particular solution, especially if it involves terms like $\ln x$ or powers of $x$.