Final answer:
The question involves finding coefficients for power series solutions to a differential equation. Without additional information, the coefficients a3, a6, b4, and b7 cannot be determined. The typical process includes substituting the series into the equation, matching powers of x, and solving the resulting system of equations.
Step-by-step explanation:
The student's question is asking for two linearly independent solutions to the third-order linear homogeneous differential equation y''' + 9xy = 0. The specific forms of solutions provided, y1 and y2, are power series solutions. Let's find the requested coefficients a3, a6, b4, and b7 for each series.
For y1 = 1 + a3x^3 + a6x^6 + ..., when substituted into the differential equation and after differentiating term by term, will yield a series where the coefficients associated with each power of x must be zero for the equation to hold for all x. Similarly, substituting y2 = x + b4x^4 + b7x^7 + ... will give us another series that has to satisfy the differential equation.
After substitution and differentiation, setting the coefficients of like powers of x to zero will give us a system of equations. Solving those equations will yield the values for a3, a6, b4, and b7. However, as this is an exercise in power series solutions, and since the question does not provide enough details to solve for the coefficients, the coefficients cannot be determined directly without additional information from the original differential equation. Typically, the process involves equating coefficients for the same powers of x and solving for the unknowns. To find the coefficients, one would usually calculate the derivatives of the assumed power series solutions, plug them into the differential equation, and match the terms of equal powers to find the relations between the coefficients.