Question

find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior for increasing\(t.\) $$ y^{\prime \prime}+y^{\prime}+1.25 y=0, \quad y(0)=3, \quad y^{\prime}(0)=1 $$

Short answer

The solution to the given second-order linear homogeneous differential equation with initial conditions is: $$ y(t) = e^{-\frac{t}{2}}(3 \cos(2t) + \frac{5}{4} \sin(2t)) $$ As time increases, the solution exhibits damped oscillatory behavior, with oscillations gradually decreasing in amplitude and approaching zero due to the damping effect of the exponential term.

Step-by-step solution

Solve the homogeneous differential equation

We have a second-order linear homogeneous differential equation, given by: $$ y^{\prime \prime} + y^{\prime} + 1.25y = 0 $$ To solve this, let's assume a solution of the form \(y(t) = e^{rt}\). Substituting this into the equation, we obtain the characteristic equation: $$ r^2 + r + 1.25 = 0 $$ We'll now solve for the roots of this quadratic equation.

Find the roots of the characteristic equation

To find the roots of the characteristic equation, we can use the quadratic formula: $$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ In our situation, \(a=1\), \(b=1\), and \(c=1.25\). Putting the values in the formula we get, $$ r = \frac{-1 \pm \sqrt{1 - 4 \cdot 1.25}}{2} = \frac{-1 \pm \sqrt{-4}}{2} $$ Since we have two complex roots, \(r_{1,2} = \frac{-1 \pm 2i}{2}\), our solution takes the form: $$ y(t) = e^{-\frac{t}{2}}(C_1 \cos(2t) + C_2 \sin(2t)) $$ Here, \(C_1\) and \(C_2\) are constants to be determined using the initial conditions.

Apply the initial conditions

The initial conditions are given by \(y(0)=3\) and \(y^{\prime}(0)=1\). Let's apply them to find the constants \(C_1\) and \(C_2\). First, using \(y(0)=3\), we get: $$ 3 = e^{0}(C_1 \cos(0) + C_2 \sin(0)) \Rightarrow C_1 = 3 $$ Now, we need to find \(y'(t)\). Differentiating \(y(t)\) with respect to \(t\), we get: $$ y^{\prime}(t) = -\frac{1}{2} e^{-\frac{t}{2}}(3 \cos(2t) + C_2 \sin(2t)) + e^{-\frac{t}{2}}(- 3 \sin(2t) + 2 C_2 \cos(2t)) $$ Now, apply the second initial condition \(y^{\prime}(0)=1\): $$ 1 = -\frac{1}{2} e^{0}(3 + 0) + e^{0}(0 + 2 C_2) \Rightarrow C_2 = \frac{5}{4} $$ So, the particular solution is: $$ y(t) = e^{-\frac{t}{2}}(3 \cos(2t) + \frac{5}{4} \sin(2t)) $$

Sketch the graph and describe the behavior

We observe that the damped oscillatory behavior of the solution is characterized by a decaying exponential envelope \(e^{-\frac{t}{2}}\), which damps the oscillations as \(t\) increases. The oscillations are sinusoidal, with a frequency of \(2\). To sketch the graph, plot the decaying exponential \(e^{-\frac{t}{2}}\) and oscillations \(3 \cos(2t) + \frac{5}{4} \sin(2t)\) together. The graph will show oscillatory behavior with an amplitude that decreases as \(t\) increases. In summary, as \(t\) increases, the solution oscillates with decreasing amplitude due to the damping effect of the exponential term, eventually approaching zero.

Key concepts

Homogeneous Differential Equation

A homogeneous differential equation is a type of differential equation in which every term is a function of the dependent variable and its derivatives. In simpler terms, there are no terms that are just functions of the independent variable alone. This means that the equation is set to zero. In our specific case, the equation is \(y'' + y' + 1.25y = 0\). This kind of equation is important in describing many physical systems where the principle of superposition applies.

• The order of the differential equation is two, as the highest derivative present is the second derivative \(y''\).
• All coefficients are constant, which classifies it as a linear homogeneous differential equation with constant coefficients.

By solving such equations, we aim to understand the behavior or state changes of systems over time.

Characteristic Equation

The characteristic equation is a tool used to solve linear homogeneous differential equations with constant coefficients. The idea is to convert the differential equation into an algebraic equation that is easier to solve.
For the differential equation given \(y'' + y' + 1.25y = 0\), substituting \(y(t) = e^{rt}\) leads us to form the characteristic equation:
\(r^2 + r + 1.25 = 0\).

• This equation is a quadratic equation, allowing us to use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.
• The coefficients \(a = 1\), \(b = 1\), \(c = 1.25\) are derived from the terms in the differential equation.

Solving this equation provides the roots that dictate the form of the general solution to the differential equation.

Complex Roots

When solving the characteristic equation, you might find complex roots, especially when dealing with second-order differential equations. These arise when the discriminant \(b^2 - 4ac\) is negative, indicating no real solutions.
In our example:

• The calculated discriminant was \(-4\), resulting in complex roots \(r_{1,2} = \frac{-1 \pm 2i}{2}\).
• The real part \(-\frac{1}{2}\) indicates the rate of exponential decay influencing the behavior of the solution over time.
• The imaginary part \(2i\) determines the oscillatory nature of the solution.

Complex roots naturally lead us to a solution characterized by exponential and sinusoidal functions, specifically \(e^{\alpha t}(C_1 \cos(\beta t) + C_2 \sin(\beta t))\), where \(\alpha\) and \(\beta\) are derived from the complex roots.

Damped Oscillation

A damped oscillation refers to an oscillatory motion where the amplitude decreases gradually over time. This phenomenon occurs in our solution due to the presence of a negative real part in the complex roots. In the solution \(y(t) = e^{-\frac{t}{2}}(3 \cos(2t) + \frac{5}{4} \sin(2t))\), the damping factor \(e^{-\frac{t}{2}}\) is responsible for the decrease in amplitude.

• The exponential term \(e^{-\frac{t}{2}}\) continuously reduces the amplitude of the oscillatory terms as time \(t\) increases.
• The remaining parts, \(\cos(2t)\) and \(\sin(2t)\), retain the oscillatory behavior, representing the sinusoidal motion.
• Damped oscillations are commonly encountered in systems where energy is gradually lost over time, such as spring-mass-damper systems.

As time progresses, the solution approaches zero, indicating that the oscillations will eventually die out, a key characteristic of damped systems.