Final Answer:
The particular solution for the differential equation x²y" - 7xy' + 15y = x, using the homogeneous solution, is y_p = x²(ln(x)/x).
Step-by-step explanation:
To find a particular solution using variation of parameters, start with the given differential equation x²y" - 7xy' + 15y = x. The associated homogeneous equation is x²y"_h - 7xy'_h + 15y_h = 0, which yields the homogeneous solution y_h = C₁x⁵ + C₂x³. For the particular solution, use the form y_p = u₁y₁ + u₂y₂, where y₁ and y₂ are linearly independent solutions of the associated homogeneous equation and u₁, u₂ are functions to be determined.
The linearly independent solutions corresponding to y₁ and y₂ are x⁵ and x³, respectively. Apply the Wronskian to find u₁ and u₂. The Wronskian W(x) = |x⁵ x³; 5x⁴ 3x²| = 2x⁷. Next, compute u₁ and u₂: u₁ = ∫(-x)/2x⁷ dx = -ln(x)/(2x⁶) and u₂ = ∫x/2x⁷ dx = ln(x)/(2x⁶).
Finally, the particular solution y_p = x²(u₁x⁵ + u₂x³) simplifies to x²(ln(x)/x). Therefore, the particular solution using variation of parameters for the given differential equation is
= x²(ln(x)/x).
This process uses the method of variation of parameters to find the particular solution of the non-homogeneous differential equation, utilizing the homogeneous solutions and the Wronskian to determine the coefficients. The resulting particular solution incorporates these coefficients into the general form, yielding the specific solution that satisfies the original non-homogeneous differential equation.