Question

Find the general solution. $$ 16 y^{(4)}-72 y^{\prime \prime}+81 y=0 $$

Short answer

Answer: The general solution for the given differential equation is $y(x) = C_1 e^{\frac{3}{2}x} + C_2 e^{-\frac{3}{2}x} + C_3 cos(\frac{3}{2}x) + C_4 sin(\frac{3}{2}x)$, where $C_1, C_2, C_3,$ and $C_4$ are arbitrary constants.

Step-by-step solution

Find the characteristic equation

First, let's find the characteristic equation. You can replace each derivative with a power of r in the following way: $$ 16r^4 - 72r^2 + 81 = 0 $$ This is the characteristic equation.

Solve for the roots of the characteristic equation

Now, let's find the roots of the characteristic equation by factoring: $$ (4r^2 - 9)(4r^2 - 9)= 0 $$ This gives us two roots: $$r_1 = \pm \dfrac{3}{2}$$ and $$r_2 = \pm \dfrac{3i}{2}$$.

Write down the general solutions

The general solution for a fourth-order linear homogeneous differential equation with distinct real and complex roots is given by: $$ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} + C_3 e^{r_3 x}cos(r_4 x) + C_4 e^{r_3 x}sin(r_4 x), $$ where $$r_1$$ and $$r_2$$ are the real roots, $$r_3$$ and $$r_4$$ are the real and imaginary parts of the complex roots, and $$C_1, C_2, C_3$$, and $$C_4$$ are arbitrary constants.

Substitute the roots and simplify the general solution

By plugging in the values we found for the roots, we obtain the general solution for the given differential equation: $$ y(x) = C_1 e^{\dfrac{3}{2}x} + C_2 e^{-\dfrac{3}{2}x} + C_3 e^0cos(\dfrac{3}{2}x) + C_4 e^0sin(\dfrac{3}{2}x) $$ Simplifying, we get: $$ y(x) = C_1 e^{\dfrac{3}{2}x} + C_2 e^{-\dfrac{3}{2}x} + C_3 cos(\dfrac{3}{2}x) + C_4 sin(\dfrac{3}{2}x) $$ This is the general solution for the given differential equation.

Key concepts

Characteristic Equation

The characteristic equation is a pivotal concept when working with linear differential equations. It provides a bridge to solving these equations by transforming the problem into an algebraic one. In our example, the given fourth-order linear homogeneous differential equation is represented as \(16 y^{(4)}-72 y''+81 y=0\). To find the characteristic equation, we substitute each derivative term with \(r^n\) where \(n\) is the order of the derivative. The resulting algebraic equation, \(16r^4 - 72r^2 + 81 = 0\), is the characteristic equation for our original differential equation.

This equation allows us to find the roots that determine the fundamental solutions to the differential equation. Solving the characteristic equation is crucial because it influences the form of the general solution we will construct, especially the type of functions involved, be they exponential, trigonometric, or a combination of these.

Fourth-Order Linear Homogeneous Differential Equation

A fourth-order linear homogeneous differential equation has the highest derivative being the fourth derivative of the unknown function. The term 'homogeneous' implies all terms of the equation involve the function or its derivatives, and the equation equals zero. The general form looks something like \(a_4y^{(4)} + a_3y^{(3)} + a_2y'' + a_1y' + a_0y = 0\) where the coefficients \(a_i\) are constants.

In solving such an equation, the characteristic equation we previously discussed becomes especially important. Once we have the roots from the characteristic equation, the general solution is a linear combination of functions that correspond to each root. If a root is real, it contributes an exponential term to the solution; if a root is complex, it contributes both an exponential and trigonometric term, reflecting the sinusoidal behavior imposed by the complex roots.

Complex Roots in Differential Equations

When solving the characteristic equation for a differential equation, complex roots often appear in conjugate pairs. If the characteristic equation has complex roots, say \(r_3 \pm i r_4\), they lead to a part of the general solution that includes both exponential and trigonometric functions, denoted typically by cosine \(\cos\) and sine \(\sin\) terms.

In our example, we find the complex roots \(\pm \dfrac{3i}{2}\), and these contribute to the general solution as \(e^{r_3 x}\cos(r_4 x)\) and \(e^{r_3 x}\sin(r_4 x)\). It's significant to note that the real part of the complex root affects the growth or decay rate of the oscillatory component, while the imaginary part determines the oscillation's frequency. The presence of both exponential and trigonometric functions in the solution to a differential equation with complex roots reflects the diverse phenomena such equations can model, including damped oscillations commonly seen in engineering and physics.