91Ó°ÊÓ

When you encounter a differential equation, a common initial strategy is to solve it using a characteristic equation. The characteristic equation is derived from assuming a solution that exhibits exponential behavior, like \( y = e^{rt} \).
By plugging this form back into the differential equation, you derive a polynomial equation in terms of \( r \).
For example, with the equation \( 4y'' - 4y' + 13y = 0 \), we substitute \( y = e^{rt} \) to get the characteristic equation:

• \( 4r^2 - 4r + 13 = 0 \)

To arrive at this, you differentiate \( y = e^{rt} \) twice and substitute these into the original differential equation, making it purely in terms of the variable \( r \).The solutions for \( r \) will often dictate the form of the general solution to the differential equation, which can be real or complex.

When solving the characteristic equation, you may find that the roots are complex numbers. Complex roots arise when the discriminant (inside the square root of the quadratic formula) is negative.
In our example, the characteristic equation is \( 4r^2 - 4r + 13 = 0 \). Solving this using the quadratic formula results in complex roots:

• \( r_1 = \frac{1}{2} + \frac{i\sqrt{3}}{2} \)
• \( r_2 = \frac{1}{2} - \frac{i\sqrt{3}}{2} \)

The general solution to a differential equation with complex roots takes the form:- \( y(t) = e^{\text{Re}(r)t} \left( C_1 \cos(\text{Im}(r)t) + C_2 \sin(\text{Im}(r)t) \right) \)
Where \( \text{Re}(r) \) is the real part and \( \text{Im}(r) \) is the imaginary part of the complex roots. This form ensures that the solution remains real-valued.

The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This is particularly useful for finding the characteristic roots in differential equations.
The formula is expressed as:

• \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Using our specific characteristic equation \( 4r^2 - 4r + 13 = 0 \), we identify \( a = 4 \), \( b = -4 \), and \( c = 13 \). Substituting these values into the quadratic formula, we find:

• \( r = \frac{4 \pm \sqrt{16 - 208}}{8} \)
• \( r = \frac{4 \pm \sqrt{-192}}{8} \)
• \( r = \frac{1 \pm i\sqrt{3}}{2} \)

The quadratic formula allows us to systematically find the roots of any quadratic, whether they are real or complex, in a reliable way. It provides a clear framework for proceeding directly from the differential equation to the general solution.