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Chapter 4: Problem 15

Classify the following ordinary differential equations by determining whether they are linear, what their order is, whether they are homogeneous, and whether their coefficients are constant. (a) \((\sin x) y^{\prime \prime}+\cos x=0\) (b) \(y^{\prime \prime}+y^{2}=2 y^{\prime}\). (c) \(\frac{d^{3} y}{d t^{3}}+\frac{2 d y}{d t}=\sin y\). (d) \(\frac{d}{d t}\left(y^{2}+2 y\right)=y\). (e) \(\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+3 y=e^{t}+e^{-t}\) (f) \(\frac{d y}{d t}=\frac{1}{1+y}\). (g) \(\frac{d y}{d x}=\frac{1}{1+x}\). (h) \(\frac{d^{5} y}{d x^{5}}=x^{6}+5 x+6\). (i) \(t \frac{d y}{d t}+t y=1\).

Short Answer

Expert verified
Equations (a) and (e) are linear. Only (e) has constant coefficients.

Step by step solution

01

- Identify the order of the differential equation

The order of a differential equation is the highest derivative present in the equation.
02

- Determine if the differential equation is linear

A differential equation is linear if the dependent variable and all its derivatives appear to the power 1 and are not multiplied by each other.
03

- Check for homogeneousness

A differential equation is homogeneous if the right-hand side is zero or can be made zero by moving terms around.
04

- Check if coefficients are constant

Check if the coefficients (the functions multiplying the derivatives or the dependent variable) are constants.
05

Classify equation (a)

For the equation \( (\sin x) y^{\prime \prime} + \cos x = 0 \): \- Order: 2 \- Linear: Yes \- Homogeneous: Yes \- Coefficients constant: No
06

Classify equation (b)

For the equation \( y^{\prime \prime} + y^{2} = 2 y^{\prime} \): \- Order: 2 \- Linear: No \- Homogeneous: Yes \- Coefficients constant: Yes
07

Classify equation (c)

For the equation \( \frac{d^{3} y}{d t^{3}} + \frac{2 d y}{d t} = \sin y \): \- Order: 3 \- Linear: No \- Homogeneous: Yes \- Coefficients constant: Yes
08

Classify equation (d)

For the equation \( \frac{d}{d t}\big( y^{2} + 2 y \big) = y \): \- Order: 1 \- Linear: No \- Homogeneous: Yes \- Coefficients constant: Yes
09

Classify equation (e)

For the equation \( \frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + 3 y = e^{t} + e^{-t} \): \- Order: 2 \- Linear: Yes \- Homogeneous: No \- Coefficients constant: Yes
10

Classify equation (f)

For the equation \( \frac{d y}{d t} = \frac{1}{1+y} \): \- Order: 1 \- Linear: No \- Homogeneous: Yes \- Coefficients constant: Yes
11

Classify equation (g)

For the equation \( \frac{d y}{d x} = \frac{1}{1+x} \): \- Order: 1 \- Linear: Yes \- Homogeneous: No \- Coefficients constant: No
12

Classify equation (h)

For the equation \( \frac{d^{5} y}{d x^{5}} = x^{6} + 5 x + 6 \): \- Order: 5 \- Linear: Yes \- Homogeneous: No \- Coefficients constant: No
13

Classify equation (i)

For the equation \( t \frac{d y}{d t} + t y = 1 \): \- Order: 1 \- Linear: Yes \- Homogeneous: No \- Coefficients constant: No

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Key Concepts

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

Order of Differential Equations
The order of a differential equation is determined by the highest derivative in the equation. For example, in the equation \((\sin x) y^{\prime \prime} + \cos x = 0\), the highest derivative present is \(y^{\prime \prime} \), which is the second derivative of \(y\). Thus, this differential equation is of the second order. Similarly, for the equation \(\frac{d^{3} y}{d t^{3}} + \frac{2 d y}{d t} = \sin y\), the highest derivative is \( \frac{d^{3} y}{d t^{3}} \), making it a third-order differential equation. Knowing the order helps in understanding the complexity and type of solutions the equation might have.
Linear Differential Equations
A differential equation is called linear if the dependent variable and all its derivatives appear to the power 1 and are not multiplied by each other. For instance, in the equation \(y^{\prime \prime} + y = 0\), both \(y\) and \(y^{\prime \prime}\) appear to the first power and are not multiplied together, so it is a linear equation. Contrarily, the equation \(y^{\prime} + y^{2} = 2 y^{\prime} \) is non-linear because \(y^{2}\) involves the dependent variable raised to the power 2. Understanding whether a differential equation is linear is crucial because linear equations generally have simpler and more predictable solutions.
Homogeneous Differential Equations
A differential equation is considered homogeneous if the free term (or the right-hand side) is zero when simplified. For example, in the equation \(y^{\prime \prime} + y = 0\), the right side is zero, making this equation homogeneous. Conversely, in the equation \(\frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + 3 y = e^{t} + e^{-t}\), the right side is a non-zero function \(e^{t} + e^{-t} \), making this equation non-homogeneous. Identifying homogeneity is important because homogeneous equations have different methods of solution than non-homogeneous ones.
Constant Coefficients in Differential Equations
The coefficients are the functions multiplying the derivatives or the dependent variable in a differential equation. If these coefficients are constants, the equation is said to have constant coefficients. For instance, in the equation \(\frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + 3 y = 0\), the coefficients are 1, 2, and 3, all constants. Conversely, in the equation \(t \frac{d y}{d t} + t y = 1\), the coefficient \(t\) changes with \(t\), so it does not have constant coefficients. Equations with constant coefficients are generally easier to solve and have well-studied solution methods.

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Most popular questions from this chapter

Show that for a decaying population $$ d N / d t=-K N \quad(K>0) $$ the time at which only half of the original population remains (the half-life) is $$ \tau_{1 / 2}=\frac{\ln 2}{K} . $$

Consider a bacterial population whose growth rate is \(d \mathbf{N} / d t=K(t) N\). Show that $$ N(t)=N_{0} \exp \left(\int_{0}^{t} K d s\right). $$

Problems 15 through 24 deal with ODEs and techniques discussed in Section \(4.8 .\) Classify the following ordinary differential equations by determining whether they are linear, what their order is, whether they are homogeneous, and whether their coefficients are constant. (a) \((\sin x) y^{\prime \prime}+\cos x=0\). (f) \(\frac{d y}{d t}=\frac{1}{1+y}\) (b) \(y^{\prime \prime}+y^{2}=2 y^{\prime}\). (g) \(\frac{d y}{d x}=\frac{1}{1+x}\). (c) \(\frac{d^{3} y}{d t^{3}}+\frac{2 d y}{d t}=\sin y\). (h) \(\frac{d^{5} y}{d x^{5}}=x^{6}+5 x+6\) (d) \(\frac{d}{d t}\left(y^{2}+2 y\right)=y\). (i) \(t \frac{d y}{d t}+t y=1\). (e) \(\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+3 y=e^{t}+e^{-t}\).

Problems 15 through 24 deal with ODEs and techniques discussed in Section \(4.8 .\) Consider the equation $$ \frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}-3 x=0 $$ (a) Show that \(x_{1}(t)=e^{3 n}\) and \(x_{2}(t)=e^{-t}\) are two solutions. (b) Show that \(x(t)=c_{1} x_{1}(t)+c_{2} x_{2}(t)\) is also a solution.

Problems 15 through 24 deal with ODEs and techniques discussed in Section \(4.8 .\) The differential equation $$ \frac{d^{2} x}{d t^{2}}+3 \frac{d x}{d t}+2 x=0 $$ has the general solution $$ x(t)=c_{1} e^{-t}+c_{2} e^{-2 t} \text {. } $$ If we are told that, when \(t=0, x(0)=1\) and its derivative \(x^{\prime}(0)=1\), we can determine \(c_{1}\) and \(c_{2}\) by solving the equations $$ \begin{aligned} &x(0)=c_{1} e^{-0}+c_{2} e^{(-2)}=c_{1}(1)+c_{2}(1)=1, \\ &x^{\prime}(0)=c_{1}(-1) e^{-6}+c_{2}(-2) e^{-0}=-c_{1}-2 c_{2}=1 . \end{aligned} $$ (a) Find the values of \(c_{1}\) and \(c_{2}\) by solving the above. For questions (b) through (e) find the solution of the differential equation subject to the specified initial condition. (b) \(y^{\prime}=10 y ; \quad y(0)=0.001\). (c) \(y^{\prime \prime}-3 y^{\prime}-4 y=0 ; \quad y(0)=0, y^{\prime}(0)=1\). (d) \(y^{\prime \prime}-9 y=0 ; \quad y(0)=5, y^{\prime}(0)=0\). (e) \(y^{\prime \prime}-5 y^{\prime}=0 ; \quad y(0)=1, y^{\prime}(0)=2\).
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