Question

Find a fundamental set of solutions. $$ (D-1)^{2}(2 D-1)^{3}\left(D^{2}+1\right) y=0 $$

Short answer

Answer: A fundamental set of solutions for the given differential equation is $\{e^t, e^{\frac{1}{2}t}, \cos{(t)}, \sin{(t)}\}$.

Step-by-step solution

1. Break down the equation

Rewriting the equation as a product of simple linear equations, we have: $$ (D-1)(D-1)(2D-1)(2D-1)(2D-1)(D^2 + 1)y = 0 $$ This gives us five linear homogeneous equations to solve: (i) $$(D-1)y = 0$$ (ii) $$(2D-1)y = 0$$ (iii) $$(D^2 + 1)y = 0$$ Note that we only need to solve equations (i), (ii), and (iii) once, as solving them again would just yield linearly dependent solutions.

2. Solve each equation

(i) Solving \((D-1)y = 0\) gives the general solution: $$ y_1 = Ce^{t} $$ (ii) Solving \((2D-1)y = 0\) gives the general solution: $$ y_2 = De^{\frac{1}{2}t} $$ (iii) Solving \((D^2 + 1)y = 0\) gives the general solution: $$ y_3 = Ee^{it} + Fe^{-it} $$ By Euler's formula, we can rewrite \(y_3\) in terms of sine and cosine functions: $$ y_3 = (E + F)\cos{(t)} + (E - F)i\sin{(t)} $$ Taking the real and imaginary parts separately, we get two independent solutions: $$ y_4 = G\cos{(t)} \quad \text{and} \quad y_5 = H\sin{(t)} $$

3. Form a fundamental set of solutions

We found five linearly independent solutions: \(y_1, y_2, y_4,\) and \(y_5\). Therefore, a fundamental set of solutions is given by: $$ \{e^t, e^{\frac{1}{2}t}, \cos{(t)}, \sin{(t)}\} $$

Key concepts

Linear Differential Equations

A linear differential equation is one of the cornerstones of calculus and applied mathematics. It involves derivatives of a function and can be classified based on its order, the highest derivative present in the equation. The primary characteristic of linear differential equations is that they can be written in the form: \[ a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_0(x)y = g(x) \]Here, each \(a_i(x)\) is a function of the independent variable \(x\), and \(g(x)\) is often zero in homogeneous equations, leading to simpler solutions.
Some key features of these equations include linearity, where solutions can often be added together to form new solutions, and they do not exhibit phenomena like chaos or complex changes in behavior, which are common in non-linear equations. Recognizing a linear differential equation is the first step in solving it.

Fundamental Set of Solutions

In the context of linear differential equations, a fundamental set of solutions is crucial. It is essentially a set of solutions that can be used to form the general solution to a differential equation.
For an \(n^{th}\) order linear differential equation, you typically need \(n\) linearly independent solutions to span the solution space completely. Each solution adds another dimension, much like how different "basis" vectors span a vector space in linear algebra.
The linear independence of the solutions is verified through the Wronskian determinant. If the Wronskian is non-zero at some point in the interval of interest, the solutions are independent and thus form a fundamental set. This implies any other solution of the differential equation can be expressed as a linear combination of the solutions in this set.

Homogeneous Equations

Homogeneous differential equations are special cases where the function \(g(x) = 0\), leading to all terms being a function of the derivatives and \(y\) itself. This characteristic implies that if \(y(x)\) is a solution, then any scalar multiple of \(y(x)\), \(cy(x)\), is also a solution, due to the equation's linear nature.
The homogeneous part of a differential equation shares the same order as the original differential equation and is set to zero. Solving it often involves finding a characteristic equation, if it has constant coefficients, or using substitution and variations of solutions for more complex cases.

• These equations are pivotal as they often form the first step in solving more complex non-homogeneous equations.
• Their importance cannot be overstated since understanding the behavior of solutions to homogeneous equations directly contributes to tackling inhomogeneous ones.

Euler's Formula

Euler's formula provides a beautiful bridge between complex exponential functions and trigonometric functions. It states that for any real number \(t\), \[ e^{it} = \cos(t) + i \sin(t) \]This identity is fundamental in converting complex solutions into real ones or separating real and imaginary parts, making them practical and interpretable in applications.
In solving differential equations, especially when dealing with complex roots of the characteristic equation, Euler's formula helps express complex exponential solutions in terms of sine and cosine functions, thereby simplifying their interpretation and integration into a fundamental set of real solutions.

• It's particularly useful when the solutions to a differential equation include imaginary components.
• Using this formula allows translating potentially complex mathematical results into real-world interpretations, especially when you require real-valued solutions.