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Chapter 31: Problem 20

Solve the given differential equations. $$2 D^{2} y-3 D y-y=0$$

Short Answer

Expert verified
The general solution is \( y(x) = C_{1} e^{\frac{3 + \sqrt{17}}{4}x} + C_{2} e^{\frac{3 - \sqrt{17}}{4}x} \).

Step by step solution

01

Understand the Form of the Differential Equation

The given differential equation is a linear homogeneous differential equation with constant coefficients of the form \(ay'' + by' + cy = 0\), where \(y''\) is the second derivative and \(D\) is the differentiation operator.
02

Write the Characteristic Equation

For the equation \(2D^{2}y - 3Dy - y = 0\), it translates to the characteristic equation \(2m^{2} - 3m - 1 = 0\), where \(m\) represents the roots corresponding to the operator \(D\).
03

Solve the Characteristic Equation

The characteristic equation \(2m^{2} - 3m - 1 = 0\) can be solved using the quadratic formula: \(m = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Here, \(a = 2\), \(b = -3\), and \(c = -1\).
04

Calculate the Discriminant

Calculate the discriminant \(b^{2} - 4ac = (-3)^{2} - 4 \cdot 2 \cdot (-1) = 9 + 8 = 17\). The discriminant is positive, indicating two distinct real roots.
05

Determine the Roots

Using the quadratic formula, the roots are \(m_{1} = \frac{3 + \sqrt{17}}{4}\) and \(m_{2} = \frac{3 - \sqrt{17}}{4}\).
06

Write the General Solution

The general solution for the differential equation with distinct real roots \(m_{1}\) and \(m_{2}\) is \(y(x) = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}\), where \(C_{1}\) and \(C_{2}\) are arbitrary constants.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Homogeneous Differential Equation
A linear homogeneous differential equation is a type of equation that involves derivatives of a function and is characterized by linearity and homogeneity. In simpler terms, these equations can be written in the structure of
  • Linear: Each term is either a constant or a product of a constant and a derivative of the function.
  • Homogeneous: The equation is set to equal zero, meaning there are no standalone constant terms.
These equations typically take the form: \[ ay'' + by' + cy = 0, \] where \(a\), \(b\), and \(c\) are constants, and \(y''\) is the second derivative of \(y\). The solution to these equations involves understanding their properties and deriving a general solution that describes the function \(y\). Such solutions are key in fields ranging from physics to engineering because they often describe dynamic systems, like oscillations or wave patterns.
Characteristic Equation
The characteristic equation is derived from a linear homogeneous differential equation by substituting the differential operator \(D\) with \(m\). This approach reduces a differential equation problem to an algebraic problem. By replacing derivatives in the equation with a characteristic polynomial, it becomes:
  • For a second-order differential equation like \(ay'' + by' + cy = 0\), the characteristic equation is: \[ am^2 + bm + c = 0, \]
  • The roots of this polynomial determine the form of the solution to the differential equation.
The significance of the characteristic equation lies in its ability to simplify solving differential equations. Depending on the solutions (roots) of the characteristic equation, the general solution of the differential equation can take several forms, aiding in understanding the behavior of the underlying system.
Quadratic Formula
The quadratic formula is a mathematical formula used to find the solutions (roots) of a quadratic equation, which is a polynomial equation of degree two. The formula is given by:\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation:\[ ax^2 + bx + c = 0. \] The quadratic formula provides a straightforward way to solve any quadratic equation by substituting in the coefficients.
  • Its main advantage is that it works universally for all quadratic equations.
  • The presence of the discriminant, \(b^2 - 4ac\), as part of the formula, offers insight into the nature of the roots, which can be real, distinct, equal, or complex.
Understanding and using the quadratic formula is crucial in differential equations, as it often forms the basis for determining the solutions to characteristic equations.
Discriminant
The discriminant is a specific part of the quadratic formula, isolated as \(b^2 - 4ac\). It provides valuable insights into the nature of the roots of the quadratic equation.
  • A positive discriminant (\(D > 0\)) indicates two distinct real roots.
  • A zero discriminant (\(D = 0\)) indicates a repeated (or double) root.
  • A negative discriminant (\(D < 0\)) indicates that the roots are complex and conjugate pairs.
Understanding the discriminant helps predict the nature of solutions without solving the entire quadratic formula. For example, in differential equations, the type of roots impacts the general solution's form - whether it will include exponential terms with real exponents or oscillatory terms due to complex roots. By analysing the discriminant, we gain pre-emptive insight into the behavior of the system described by the differential equation.

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Most popular questions from this chapter

Solve the given differential equations. $$D^{2} y+5 D y+4 y=x e^{x}+4$$

Find the inverse transforms of the given functions of \(s\) $$F(s)=\frac{2 s+3}{s^{2}-2 s+5}$$

Solve the given problems. For a given circuit, \(L=0.100 \mathrm{H}, R=0, C=100 \mu \mathrm{F},\) and \(E=100 \mathrm{V} .\) Find the equation relating the charge and the time if \(q=0\) and \(i=0\) when \(t=0\)

Find the inverse transforms of the given functions of \(s\) $$F(s)=\frac{s+3}{s^{2}+4 s+13}$$

Find the simplest form of the second-order homogeneous linear differential equation that has the given solution. $$y=c_{1} e^{3 x}+c_{2} e^{-3 x}$$
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