Question

Find the general solution. $$ 4 y^{(4)}+12 y^{\prime \prime \prime}+3 y^{\prime \prime}-13 y^{\prime}-6 y=0 $$

Short answer

Answer: The general solution is \(y(x) = C_1e^{-1.172x} + C_2e^{-0.614x} + C_3e^{0.393x} + C_4e^{2.392x}\), where \(C_1, C_2, C_3,\) and \(C_4\) are constants.

Step-by-step solution

Write down the characteristic equation

We will write down the characteristic equation associated with the given linear differential equation. We do this by substituting \(y = e^{rx}\), where \(r\) is a characteristic root. The characteristic equation for this differential equation is: $$ 4 r^4 + 12 r^3 + 3 r^2 - 13 r - 6 = 0 $$

Solve the characteristic equation

Next, we need to find the roots of the characteristic equation. In this case, finding the roots analytically isn't feasible, so we will use a numerical method or a calculator to find the approximate roots: $$ r \approx -1.172, -0.614, 0.393, 2.392 $$ These roots are our characteristic roots.

Construct the general solution

Now that we have the roots, we can construct the general solution. Since all the roots are distinct, the general solution will be in the form: $$ y(x) = C_1e^{r_1x} + C_2e^{r_2x} + C_3e^{r_3x} + C_4e^{r_4x} $$ Substituting the roots we found earlier: $$ y(x) = C_1e^{-1.172x} + C_2e^{-0.614x} + C_3e^{0.393x} + C_4e^{2.392x} $$ This is the general solution of the given homogeneous linear differential equation, where \(C_1, C_2, C_3,\) and \(C_4\) are constants to be determined by boundary or initial conditions if provided.

Key concepts

Characteristic Equation

A characteristic equation is crucial in solving higher order differential equations. It helps us transition from the differential equation into a polynomial equation. This is achieved by assuming solutions of the form \(y = e^{rx}\), where \(r\) represents the characteristic roots or values. By substituting \(y = e^{rx}\) into the differential equation, each term involving a derivative of \(y\) can be expressed simply in terms of \(r\).

For example, the fourth derivative of \(y\) with respect to \(x\) will transform into \(r^4e^{rx}\), making the equation entirely in terms of \(r\).

• This transition shrinks the problem into solving a polynomial equation known as the characteristic equation.
• Here, the polynomial results from coefficients and powers of \(r\) derived from the original differential equation.

Finding the roots of this polynomial is essential. These roots directly inform us how to frame the general solution of the differential equation.

General Solution

The general solution of a differential equation comprises all possible solutions. Once we solve the characteristic equation and obtain the roots, these roots serve as the exponents in terms like \(e^{rx}\) in the solution.

If you have \(n\) distinct real roots, the general solution forms a linear combination of terms with these roots as exponents. So for four distinct roots, \(r_1, r_2, r_3, r_4\), the general solution takes the form: \[ y(x) = C_1e^{r_1x} + C_2e^{r_2x} + C_3e^{r_3x} + C_4e^{r_4x} \] Here, \(C_1, C_2, C_3\), and \(C_4\) are constants which can be determined through initial or boundary conditions.

• These constants represent the arbitrary nature of the solution, accounting for infinite possibilities without specific conditions.
• Solving for \(C_1, C_2, C_3, C_4\) requires additional information, such as initial values.

The power of the general solution lies in its ability to describe the family of all possible solutions for the differential equation.

Characteristic Roots

Characteristic roots stem from the roots of the characteristic equation. They are pivotal in constructing the solution of a differential equation.

These are the values of \(r\) that satisfy the characteristic equation formed from the original differential equation. Depending on their nature, the characteristic roots can be real and distinct, real and repeated, or complex conjugates.

• Real and distinct roots directly lead to individual exponential solutions of the form \(e^{rx}\).
• If the roots are real and repeated, terms from \(e^{rx}\) to \(xe^{rx}\), and further polynomial terms with \(x\) in front, can be involved.
• If the roots are complex, their real components contribute to exponential decay/growth, and their imaginary parts lead to oscillatory solutions.

The distinct roots obtained from our particular case show that each leads to a unique exponential function contributing to the general solution.

Homogeneous Linear Differential Equation

A homogeneous linear differential equation is one where all terms are dependent on the function and its derivatives, with no standalone constants or functions of the independent variable present. This means the equation equates to zero.

These equations have critical properties that simplify solving them, especially the linearity and homogeneity.

• Linearity allows us to use superposition; solutions can be added together to form new solutions.
• Homogeneity implies that solutions are purely based on the structure of the differential equation, without external forcing.

The absence of non-homogeneous parts simplifies finding the characteristic equation and, subsequently, the characteristic roots. These roots will guide us in forming the general solution for the equation. In short, the homogeneous linear differential equation is an elegant form that aids in comprehending the behavior and solutions of higher order differential equations.