Question

For each of the following differential equations with associated initial conditions, find the solution on the interval \([0, \infty)\). (a) \(y^{\prime \prime}(t)+4 y^{\prime}(t)+7 y(t)=u_{1}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (b) \(y^{\prime \prime}(t)+2 y^{\prime}(t)+3 y(t)=\delta_{\pi}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (c) \(y^{\prime \prime}(t)-2 y^{\prime}(t)+y(t)=(-1)^{[t]}, \quad y(0)=0, \quad y^{\prime}(0)=0\) (d) \(y^{\prime \prime}(t)+2 y^{\prime}(t)+2 y(t)=\delta_{\pi}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (e) \(y^{\prime \prime}(t)+4 y(t)=\delta_{\pi}(t)-\delta_{2 \pi}(t), \quad y(0)=0, \quad y^{\prime}(0)=0\) (f) \(y^{\prime \prime \prime}(t)-y(t)=\left\\{\begin{array}{ll}1, & \pi \leq t \leq 2 \pi, \\ 0, & \text { otherwise, }\end{array}\right\\} \quad y(0)=y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1\)

Short answer

Apply methods like characteristic equations and Laplace transforms to solve differential equations for cases (a)-(f), using initial conditions to find exact solutions.

Step-by-step solution

Classify the Equation Type

The equations presented are linear differential equations with specific initial conditions. Before solving, identify if each is homogeneous or non-homogeneous based on the presence of external functions such as the unit and delta impulses.

Solve the Homogeneous Equation

For each equation, solve the associated homogeneous differential equation. This involves finding the characteristic equation and its roots. Use these roots to write the general solution for the homogeneous differential equation.

Solve the Particular Solution

Depending on the nature of the non-homogeneous part, use methods such as undetermined coefficients or Laplace transforms to find a particular solution that satisfies the non-homogeneous part of the equation.

Apply Initial Conditions

Combine the homogeneous and particular solutions, then apply the initial conditions to find specific constants. This provides the complete specific solution for each differential equation.

Analyze Each Case

For each case (a)-(f), carefully perform the above steps. Note that cases involving delta functions or piecewise functions may require the use of Laplace transforms and discontinuity handling methods.

Key concepts

Initial Conditions

Initial conditions are specific values assigned to a solution of a differential equation at a certain point, usually the beginning of the interval under consideration. They are a necessary part of solving ordinary differential equations because they allow us to determine the particular solution that fits a given problem. For each differential equation in this exercise, the initial conditions include values at time zero, such as \(y(0)\) and \(y'(0)\).
The role of initial conditions is crucial because:

• They help define a unique solution from a family of possible solutions.
• They ensure continuity and smoothness of the solution within the given interval.
• Initial conditions are particularly needed when solving physical problems where solutions must satisfy certain state conditions at a specific time.

To apply initial conditions, we combine them with both the homogeneous and particular solutions of the differential equation to solve for any unknown constants.

Homogeneous Equations

Homogeneous equations are differential equations where the sum of functions equals zero. The general form does not include any external forcing functions like unit step functions or delta impulses. Solving homogeneous equations involves finding the roots of the characteristic equation derived from the differential equation.
Key steps to solve homogeneous equations include:

• Formulate the characteristic equation from the given differential equation.
• Solve for the roots, which could be real, complex, or repeated.
• Develop the general solution using these roots. For real and distinct roots \(r_1\) and \(r_2\), the solution is \(y_h(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\).
• If the roots are complex, they will be in the form \(a \pm bi\), providing solutions featuring sine and cosine functions.

The homogeneous solution represents how the system behaves without external disturbances.

Non-Homogeneous Equations

Unlike homogeneous equations, non-homogeneous differential equations include an external function or forcing term, which could be constant, variable, or an impulse like a delta function. These additional terms modify the behavior of the system beyond its natural state.
Approach to solve non-homogeneous equations:

• First, solve the homogeneous part to get the natural behavior of the system.
• Identify a particular solution that fits the external forcing term. The method of undetermined coefficients or variation of parameters are common techniques. Laplace transforms can be particularly useful for impulse functions.
• Add the particular solution to the homogeneous solution to determine the general solution.
• Apply initial conditions to find the complete specific solution.

Solving for the particular solution requires creativity and sometimes trial and error, especially when handling piecewise or impulsive functions.

Laplace Transforms

Laplace transforms are a valuable tool in solving linear differential equations, especially when dealing with non-homogeneous equations involving impulse functions like the delta function. They transform differential equations into algebraic equations, which are often easier to solve.
The process involves:

• Taking the Laplace transform of the entire differential equation, including initial conditions, which simplifies handling of derivatives.
• Solving the resulting algebraic equation.
• Applying the inverse Laplace transform to return to the time domain and obtain the solution in its natural setting.
• Handling complex initial-value problems or piecewise functions effectively through properties like linearity and shifting theorems.

In this exercise, Laplace transforms help manage delta impulses and step functions efficiently, providing insights into how systems respond over time.