Question

Find a fundamental set of solutions. $$ \left(D^{2}+9\right)^{3} D^{2} y=0 $$

Short answer

Question: Determine the fundamental set of solutions for the higher order differential equation \((D^2 + 9)^3 D^2 y = 0\). Answer: The fundamental set of solutions for the given differential equation is $\{c_1, c_2x, \cos(3x), \sin(3x), \cos(x), \sin(x)\}$.

Step-by-step solution

Factor the differential operator

Factor the differential operator as \((D^2 + 9)^3D^2\). Since the differential equation is linear and homogeneous, we can consider these parts separately.

Solve for each part

First, let's solve each part for the simpler equation \(D^2 y = 0\). This is a second-order homogeneous linear differential equation with constant coefficients. The characteristic equation is \(r^2=0\), which has a repeated root \(r=0\). Therefore, the general solution for this part is \(y=c_1+c_2x\). Next, we solve the equation \((D^2+9)^3y=0\). This equation can be rewritten as \((D^2+9)^3y=0\). Let \(y=e^{rx}\). Then, the equation becomes \(((r^2 + 9)^3)e^{rx}=0\). Since the exponential factor cannot be zero, we have \((r^2+9)^3=0\) which gives us complex roots of the form \(r=\pm 3i, \pm i\). For each pair of complex roots, we can obtain two independent solutions by taking the real and imaginary parts of the complex solutions. In this case, the general solutions corresponding to the complex roots are as follows: - For \(r=3i\): \(e^{3ix}=\cos(3x)+i\sin(3x)\), which gives the real and imaginary parts as independent solutions: \(y_3=\cos(3x)\) and \(y_4=\sin(3x)\). - For \(r=i\): \(e^{ix}=\cos(x)+i\sin(x)\), which gives the real and imaginary parts as independent solutions: \(y_5=\cos(x)\) and \(y_6=\sin(x)\).

Combine general solutions to form the fundamental set

Now we can form a fundamental set of solutions by combining the general solutions obtained in Steps 2, which are \(y_1=c_1\), \(y_2=c_2x\), \(y_3=\cos(3x)\), \(y_4=\sin(3x)\), \(y_5=\cos(x)\), and \(y_6=\sin(x)\). The fundamental set of solutions is therefore: $$\{c_1, c_2x, \cos(3x), \sin(3x), \cos(x), \sin(x)\}$$

Key concepts

Differential Operator

A differential operator is a tool used in calculus that involves derivatives. Imagine it as an instruction telling you how to differentiate a function. In the equation \((D^2 + 9)^3 D^2 y = 0\), the letter \(D\) stands for differentiation with respect to \(x\).

Here's what it does:

• \(D^2\) means "differentiate twice." If applied to a function \(y(x)\), it gives \(\frac{d^2y}{dx^2}\).
• The expression \((D^2 + 9)\) means "apply \(D^2\) to \(y\) and then add 9 times \(y\) back to the result."

Differential operators help simplify complex equations by breaking them into manageable parts.

Understanding how these operators work is crucial for solving differential equations, especially when tackling higher-order equations like in our exercise.

Homogeneous Linear Differential Equation

A homogeneous linear differential equation is one where every term is a function of \(y\) and its derivatives, and there are no extra functions or constants added.

In our exercise, the equation is everything on the left side of \(= 0\). The "homogeneous" part indicates that the equation equals zero, meaning the solutions represent equilibrium states or steady conditions.

The linearity implies that if \(y_1\) and \(y_2\) are solutions, then any combination \(c_1y_1 + c_2y_2\) is also a solution.

• This property is useful because it allows us to find families of solutions.

This creates a "superposition" effect, which simplifies the structure of the solution and helps in creating the fundamental set of solutions.

Characteristic Equation

The characteristic equation is a fundamental tool for finding solutions to differential equations with constant coefficients. It transforms a differential equation into an algebraic one, which is often easier to solve.

For a term like \(D^2 y = 0\), the characteristic equation is \(r^2 = 0\). Solving for \(r\) gives the roots of the equation, which help determine the form of the solution.

In the exercise:

• The root \(r = 0\) indicates repeated solutions like constants \(c_1\) and linear terms \(c_2x\).
• With \((r^2 + 9)^3 = 0\), the roots \(r = \pm 3i\) and \(\pm i\) yield complex solutions.

Understanding how to derive and solve these characteristic equations is key to determining the correct form of the solutions and completing the exercise effectively.

Complex Roots

Complex roots occur when solving a characteristic equation results in solutions that include imaginary numbers. They often appear in equations with oscillatory behavior.

In the exercise, complex roots \(r = \pm 3i\) and \(\pm i\) indicate that the solutions to the differential equation will be trigonometric functions like sines and cosines.

Here's how it works:

• Complex roots \(a \pm bi\) lead to solutions: \(e^{ax}(C_1 \cos(bx) + C_2 \sin(bx))\).
• This pattern generates paired solutions that naturally account for the real and imaginary parts of complex numbers.

By understanding complex roots, we can identify key aspects of the behavior of the solutions, leveraging their oscillatory nature to form the complete solution set efficiently.