91Ó°ÊÓ

To solve a differential equation of the form \(w'' + 3xw' - w = 0\), we often begin by expressing it as a characteristic equation. This step is crucial for finding the function's specific form. The characteristic equation is derived from the differential equation by assuming a solution of the type \(w(x) = e^{mx}\), where \(m\) represents a constant. This assumption converts the differential equation into an algebraic equation in \(m\), often called the characteristic equation. For our problem, the characteristic equation is \(m^2 + 3xm - 1 = 0\). Here, \(m^2\) corresponds to \(w''\), \(3xm\) corresponds to \(3xw'\), and \(-1\) is due to \(-w\). Being able to set up the characteristic equation correctly is vital for the later steps where solutions and more specific functions are derived.

The quadratic formula is a powerful tool for solving equations of the form \(ax^2 + bx + c = 0\). It is represented as \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In our characteristic equation \(m^2 + 3xm - 1 = 0\), we apply the quadratic formula to find the roots. The values \(a = 1\), \(b = 3x\), and \(c = -1\) are identified from the characteristic equation. By substituting these into the quadratic formula, we get:

• \(b^2 - 4ac = (3x)^2 - 4(1)(-1) = 9x^2 + 4\)
• \(m = \frac{-3x \pm \sqrt{9x^2 + 4}}{2}\)

The result consists of two roots, \(m_1\) and \(m_2\), which are crucial in forming the general solution of the differential equation. Calculating these roots accurately sets the stage for deriving a well-defined solution.

Initial conditions are values given for a function and its derivatives at a specific point, which allow us to find particular solutions to differential equations. In our exercise, the initial conditions are \(w(0) = 2\) and \(w'(0) = 0\). These conditions help us solve for the constants \(A\) and \(B\) in the general solution \(w(x) = A \cdot e^{m1 \cdot x} + B \cdot e^{m2 \cdot x}\). To apply initial conditions, substitute \(x = 0\) into both equations:

• \(w(0) = A + B = 2\)
• \(w'(0) = (Am_1 + Bm_2) = 0\)

Solving these simultaneous equations will yield the values for \(A\) and \(B\), providing the unique solution to the differential equation that satisfies the initial conditions. This process ensures the solution behaves exactly as specified by the conditions set at the outset.