Problem 14 - A First Course in Differential Equations with Modeling Applications

Question

Find the general solution of the given second-order differential equation. $$2 y^{\prime \prime}-3 y^{\prime}+4 y=0$$

Step-by-Step Solution

Answer

The general solution is \(y(t) = e^{\frac{3}{4} t}(c_1 \cos(\frac{\sqrt{23}}{4} t) + c_2 \sin(\frac{\sqrt{23}}{4} t))\).

Step 1: Identify the Type of Differential Equation

The given differential equation is a second-order linear homogeneous differential equation with constant coefficients. It has the form \(2y^{\prime\prime} - 3y^{\prime} + 4y = 0\).

Step 2: Write the Characteristic Equation

For a second-order linear differential equation of the form \(ay^{\prime\prime} + by^{\prime} + cy = 0\), the characteristic equation is \(ar^2 + br + c = 0\). Substitute \(a = 2\), \(b = -3\), and \(c = 4\) into the characteristic equation:\[2r^2 - 3r + 4 = 0\]

Step 3: Solve the Characteristic Equation

Use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(r\):\[\begin{align*}a &= 2, \quad b = -3, \quad c = 4, \r &= \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 2 \times 4}}{2 \times 2} \&= \frac{3 \pm \sqrt{9 - 32}}{4} \&= \frac{3 \pm \sqrt{-23}}{4} \&= \frac{3 \pm i\sqrt{23}}{4}.\end{align*}\]

Step 4: Write the General Solution

Since the roots are complex \(r = \frac{3}{4} \pm i\frac{\sqrt{23}}{4}\), the general solution takes the form:\[y(t) = e^{\alpha t}(c_1 \cos(\beta t) + c_2 \sin(\beta t)),\]where \(\alpha = \frac{3}{4}\) and \(\beta = \frac{\sqrt{23}}{4}\). Thus, the general solution is:\[y(t) = e^{\frac{3}{4} t}(c_1 \cos(\frac{\sqrt{23}}{4} t) + c_2 \sin(\frac{\sqrt{23}}{4} t))\].

Key Concepts

Characteristic Equation

When solving second-order linear homogeneous differential equations with constant coefficients, a key step is forming the characteristic equation. For a given differential equation of the form \(ay^{\prime\prime} + by^{\prime} + cy = 0\), we replace the derivatives with powers of \(r\) to formulate the characteristic equation, \(ar^2 + br + c = 0\). This characteristic equation is a quadratic equation in terms of \(r\), a variable that helps us find the nature of the solutions. Once we derive the characteristic equation, we can apply methods, such as factoring or using the quadratic formula, to find the roots which guide us in constructing the general solution.

Complex Roots

Upon solving the characteristic equation, it's possible to encounter complex roots. Complex roots occur when the discriminant \(b^2 - 4ac\) of the characteristic equation is negative. These roots take the form \(r = \alpha \pm i\beta\), where \(\alpha\) and \(\beta\) are real numbers, and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\). These roots indicate that the solution will include sinusoidal functions. Specifically, the general solution involves the exponential function along with sine and cosine terms, capturing oscillatory behavior. Recognizing complex roots is essential, as they show that the system modeled by the differential equation has periodic characteristics.

General Solution

The general solution of a second-order differential equation with complex roots reflects both exponential growth and oscillatory nature. If the roots from the characteristic equation are \(r = \alpha \pm i\beta\), the solution takes the form \(y(t) = e^{\alpha t}(c_1 \cos(\beta t) + c_2 \sin(\beta t))\). In this equation:

• \(e^{\alpha t}\) represents an exponential growth or decay, depending on the sign of \(\alpha\).
• \(c_1\) and \(c_2\) are constants determined by initial conditions.
• \(\cos(\beta t)\) and \(\sin(\beta t)\) represent oscillations due to the imaginary component \(i\beta\).

This structure of the general solution shows how incorporating complex roots leads to solutions that blend exponential and sinusoidal functions, resonating with the dynamic behavior of many physical systems.

Homogeneous Differential Equations

A differential equation is called homogeneous if it can be expressed in the form \(ay^{\prime\prime} + by^{\prime} + cy = 0\), meaning there are no terms that are just functions of the independent variable (e.g., the variable \(t\) in this context). Homogeneous differential equations describe systems where the solution depends solely on the system's inherent properties and initial conditions. These types of equations typically model scenarios where forces are in balance, like undamped oscillations, or generally where there is no input forcing the system from outside. Homogeneous differential equations with constant coefficients are frequently found in physics and engineering as they elegantly describe relationships within systems devoid of external influences.