Question

Find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as \(t\) increases. $$ y^{\prime \prime}+4 y^{\prime}+3 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-1 $$

Short answer

Answer: The specific solution to the initial value problem is \(y(t) = 5 e^{-t} - 3 e^{-3t}\).

Step-by-step solution

Write down the initial value problem

The given initial value problem is: $$ y^{\prime \prime}+4 y^{\prime}+3 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-1 $$

Find the characteristic equation

The characteristic equation associated with the given homogeneous differential equation is: $$ r^2 + 4r + 3 = 0 $$

Solve the characteristic equation

Solve the quadratic equation by factoring it: $$ (r + 3)(r + 1) = 0 $$ The roots of the equation are \(r_1 = -1\) and \(r_2 = -3\).

Construct the general solution

Using the roots, we can construct the general solution of the homogeneous differential equation: $$ y(t) = C_1 e^{-t} + C_2 e^{-3t} $$

Apply the initial conditions

Next, we apply the initial conditions to find the specific solution. First, applying the initial condition \(y(0) = 2\), we get: $$ 2 = C_1 e^{0} + C_2 e^{0} \Rightarrow 2 = C_1 + C_2 $$ Now, applying \(y^{\prime}(0)=-1\), we differentiate the general solution and evaluate it at \(t=0\): $$ y^{\prime}(t) = -C_1 e^{-t} - 3C_2 e^{-3t} $$ $$ -1 = -C_1 e^{0} - 3C_2 e^{0} \Rightarrow -1 = -C_1 - 3C_2 $$ Solve the system of linear equations to find \(C_1\) and \(C_2\): $$ C_1 = 5, \quad C_2 = -3 $$ Now, the specific solution is: $$ y(t) = 5 e^{-t} - 3 e^{-3t} $$

Describe the behavior of the solution as \(t\) increases

As \(t\) increases, both terms in the solution decrease due to the exponential decay of the terms. The term with the faster decay is the one with \(e^{-3t}\), so it will approach zero more quickly than the term with \(e^{-t}\). As a result, the behavior of the solution will be dominated by the slower decaying term (\(5e^{-t}\)) as \(t\) increases. Overall, this solution represents a decaying function that approaches zero as \(t\) goes to infinity.