72.1 Nonhomogeneous Differential Equations Concept Overview
NONHOMOGENEOUS DIFFERENTIAL EQUATIONS | CONCEPT OVERVIEW The topic of NONHOMOGENEOUS DIFFERENTIAL EQUATIONS can be referenced on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: A DIFFERENTIAL EQUATION is considered HIGHER ORDER and HOMOGENEOUS if all the terms contain the DEPENDENT VARIABLE or an associated DERIVATIVE to an ORDER higher than one. This same DIFFERENTIAL EQUATION can become of the NONHOMOGENEOUS form if the sum DOES NOT equal zero, but rather a nonzero forcing function of the INDEPENDENT VARIABLE. In formulaic terms, a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION can generally be written as: π! π¦ ππ¦ + π + ππ¦ = π(π₯) ππ₯ ! ππ₯ Or: π¦ !! + ππ¦β² + ππ¦ = π(π₯)
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Both of these expressions are equivalent. Illustrating this further, taking the original general formula above, we turned: π! π¦ ππ¦ + π + ππ¦ = π(π₯) ππ₯ ! ππ₯ In to: π¦ !! + ππ¦β² + ππ¦ = π(π₯) By replacing: π! π¦ = π¦β²β² ππ₯ ! ππ¦ = π¦β² ππ₯ Thatβs all, nothing complex to it, however, you may see these types of equations presented in both ways, so be comfortable with each. The GENERAL FORM of the COMPLETE SOLUTION of a DIFFERENTIAL EQUATION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The GENERAL SOLUTION, otherwise referred to as the COMPLETE SOLUTION, of a DIFFERENTIAL EQUATION is expressed as: π¦ π₯ = π¦! π₯ + π¦! (π₯) Where: β’ π¦! (π₯) is the PARTICULAR SOLUTION with π π₯ present β’ π¦! (π₯) is the HOMOGENEOUS SOLUTION corresponding to π π₯ = 0 It is important to always remember that the GENERAL SOLUTION for a NONHOMOGENEOUS DIFFERENTIAL EQUATION will be made up these two unique components, the PARTICULAR SOLUTION, which may also be referred to as the NONHOMOGENEOUS SOLUTION, and the HOMEGENEOUS SOLUTION. HOMOGENEOUS SOLUTION ππ π : The STANDARD FORM for the HOMOGENEOUS SOLUTION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When we are determining the HOMOGENEOUS SOLUTION of a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION, we will ignore the DRIVING FUNCTION, g(x), on the right side of the equation and solve it as if it was HOMOGENEOUS.
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We will employ the same process that we learned in our review on HIGHER ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS. As a quick review, we will proceed as follows: 1. Define the CHARACTERISTIC POLYNOMIAL (also known as the CHARACTERISTIC EQUATION) 2. Determine the ROOTS 3. Classify the ROOTS as: a. REAL and DISTINCT b. REAL and EQUAL c. COMPLEX 4. Write the GENERAL SOLUTION based on the CLASSIFICATION of the ROOTS PARTICULAR SOLUTION ππ π : The PARTICULAR SOLUTION of a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION is defined by using the METHOD OF UNDETERMINED COEFFICIENTS. This is a method, which in practice, generally requires that we βguessβ the PARTICULAR SOLUTION based on the DRIVING FUNCTION on the right side of the equation.
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We then will take this βguessβ, DIFFERENTIATE it a few times, and plug those results back in to the left side of the equation in place of the original DERIVATIVE forms. At this point, we will take this new formula, combine the like terms to simplify it, and then set the UNDETERMINED COEFFICIENTS (which can also be thought simply as UNKNOWN COEFFICIENTS) on the left side of the equation to their corresponding values on the right side of the equation. This will provide us a SET OF EQUATIONS that we can then proceed in solving for each UNKNOWN, defining our PARTICULAR SOLUTION. Fortunately, on this exam, there will be no βguessingβ needed as we are provided a TABLE highlighting the various FORMS of a PARTICULAR SOLUTION based on the corresponding DRIVING FUNCTION. This TABLE can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A few specific forms of the function π(π₯), as it is being referred to as in the NCEES Reference Handbook, but we are using π(π₯) for clarity purposes, and itβs resulting PARTICULAR SOLUTION, π¦! (π₯), are defined as:
π(π₯)
π¦! (π₯)
π΄
π΅
π΄π !"
π΅π !" ; πΌ β π!
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π΄! sin ππ₯ + π΄! cos ππ₯
π΅! sin ππ₯ + π΅! cos ππ₯
While we are provided this table in the reference handbook, it is often not clear what the particular form of the solution should be. Here are a few guidelines that are not in the NCEES REFERENCE HANDBOOK, but may guide you when working problems involving the METHOD OF UNDETERMINED COEFFICIENTS: β’ If f x is constant, then y! = Cx β’ If f x is linear, then y! = B! x + B! β’ If f x is quadratic, then y! = Ax ! + Bx + C β’ If f x features e!" , then y! = Be!" β’ If f x features sin cx or cos cx , then y! = B! sin cx + B! cos(cx) Note that if the driving function π π₯ , or π π₯ in our teaching approach, is π !" and the term π !" also appears in the HOMOGENEOUS SOLUTION, the PARTICULAR SOLUTION will take the form π΄π₯π !" . The take away from the table is to match the DRIVING FUNCTION in the left column with the PARTICULAR SOLUTION function in the right column.
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To find the values of the UNDETERMINED COEFFICIENTS, substitute the assumed form of π¦! (π₯) and itβs associated DERIVATIVES in to the differential equation and equate the UNKNOWN COEFFICIENTS based on the values that correspond to those terms on the opposite side of the equation. The GENERAL FORM of the COMPLETE SOLUTION of a DIFFERENTIAL EQUATION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. It is important to note once again that the GENERAL SOLUTION for a NONHOMOGENEOUS DIFFERENTIAL EQUATION will always be made up these two unique components, the PARTICULAR SOLUTION, which may also be referred to as the NONHOMOGENEOUS SOLUTION, and the HOMEGENEOUS SOLUTION. Once they are both defined and placed together, they will represent the COMPLETE SOLUTION. The GENERAL SOLUTION, otherwise referred to as the COMPLETE SOLUTION, of a DIFFERENTIAL EQUATION is expressed as: π¦ π₯ = π¦! π₯ + π¦! (π₯) Where: β’ π¦! (π₯) is the PARTICULAR SOLUTION with π π₯ present β’ π¦! (π₯) is the HOMOGENEOUS SOLUTION corresponding to π π₯ = 0
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Once this COMPLETE SOLUTION is defined, INITIAL CONDITIONS can be used to evaluate any further unknown coefficients that remain from the HOMOEGENOUS portion of this solution.
FIRST-ORDER INSTRUMENTS: The GENERAL FORMULA for the DIFFERENTIAL EQUATION of FIRST-ORDER INSTRUMENTS can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A FIRST ORDER LINEAR INSTRUMENT has an OUTPUT which is given by a NONHOMOGENEOUS FIRST ORDER LINEAR DIFFERENTIAL EQUATION, which is generally written as:
π
ππ¦ + π¦ = πΎπ₯ π‘ ππ‘
Where: β’ π is the time constant of the instrument, which is a measure of the time delay in the response to changes of input. The time constant is given in units of time β’ π¦(π‘) is the response of the system output to the forcing function β’ π₯(π‘) is the forcing function β’ πΎ is the gain of the system
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The INITIAL CONDITION for this first order system is given as: π¦ 0 = πΎπ΄ Thermometers measuring temperature are first order instruments. The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured. A first order system subjected to a constant force applied instantaneously at the initial time π‘ = 0 is represented by the step function:
π₯ π‘ =
π΄ π‘
CONCEPT INTRO: A DIFFERENTIAL EQUATION is considered HIGHER ORDER and HOMOGENEOUS if all the terms contain the DEPENDENT VARIABLE or an associated DERIVATIVE to an ORDER higher than one. This same DIFFERENTIAL EQUATION can become of the NONHOMOGENEOUS form if the sum DOES NOT equal zero, but rather a nonzero forcing function of the INDEPENDENT VARIABLE. In formulaic terms, a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION can generally be written as: π! π¦ ππ¦ + π + ππ¦ = π(π₯) ππ₯ ! ππ₯ Or: π¦ !! + ππ¦β² + ππ¦ = π(π₯)
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Both of these expressions are equivalent. Illustrating this further, taking the original general formula above, we turned: π! π¦ ππ¦ + π + ππ¦ = π(π₯) ππ₯ ! ππ₯ In to: π¦ !! + ππ¦β² + ππ¦ = π(π₯) By replacing: π! π¦ = π¦β²β² ππ₯ ! ππ¦ = π¦β² ππ₯ Thatβs all, nothing complex to it, however, you may see these types of equations presented in both ways, so be comfortable with each. The GENERAL FORM of the COMPLETE SOLUTION of a DIFFERENTIAL EQUATION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The GENERAL SOLUTION, otherwise referred to as the COMPLETE SOLUTION, of a DIFFERENTIAL EQUATION is expressed as: π¦ π₯ = π¦! π₯ + π¦! (π₯) Where: β’ π¦! (π₯) is the PARTICULAR SOLUTION with π π₯ present β’ π¦! (π₯) is the HOMOGENEOUS SOLUTION corresponding to π π₯ = 0 It is important to always remember that the GENERAL SOLUTION for a NONHOMOGENEOUS DIFFERENTIAL EQUATION will be made up these two unique components, the PARTICULAR SOLUTION, which may also be referred to as the NONHOMOGENEOUS SOLUTION, and the HOMEGENEOUS SOLUTION. HOMOGENEOUS SOLUTION ππ π : The STANDARD FORM for the HOMOGENEOUS SOLUTION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When we are determining the HOMOGENEOUS SOLUTION of a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION, we will ignore the DRIVING FUNCTION, g(x), on the right side of the equation and solve it as if it was HOMOGENEOUS.
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We will employ the same process that we learned in our review on HIGHER ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS. As a quick review, we will proceed as follows: 1. Define the CHARACTERISTIC POLYNOMIAL (also known as the CHARACTERISTIC EQUATION) 2. Determine the ROOTS 3. Classify the ROOTS as: a. REAL and DISTINCT b. REAL and EQUAL c. COMPLEX 4. Write the GENERAL SOLUTION based on the CLASSIFICATION of the ROOTS PARTICULAR SOLUTION ππ π : The PARTICULAR SOLUTION of a HIGHER ORDER NONHOMOGENEOUS DIFFERENTIAL EQUATION is defined by using the METHOD OF UNDETERMINED COEFFICIENTS. This is a method, which in practice, generally requires that we βguessβ the PARTICULAR SOLUTION based on the DRIVING FUNCTION on the right side of the equation.
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We then will take this βguessβ, DIFFERENTIATE it a few times, and plug those results back in to the left side of the equation in place of the original DERIVATIVE forms. At this point, we will take this new formula, combine the like terms to simplify it, and then set the UNDETERMINED COEFFICIENTS (which can also be thought simply as UNKNOWN COEFFICIENTS) on the left side of the equation to their corresponding values on the right side of the equation. This will provide us a SET OF EQUATIONS that we can then proceed in solving for each UNKNOWN, defining our PARTICULAR SOLUTION. Fortunately, on this exam, there will be no βguessingβ needed as we are provided a TABLE highlighting the various FORMS of a PARTICULAR SOLUTION based on the corresponding DRIVING FUNCTION. This TABLE can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A few specific forms of the function π(π₯), as it is being referred to as in the NCEES Reference Handbook, but we are using π(π₯) for clarity purposes, and itβs resulting PARTICULAR SOLUTION, π¦! (π₯), are defined as:
π(π₯)
π¦! (π₯)
π΄
π΅
π΄π !"
π΅π !" ; πΌ β π!
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π΄! sin ππ₯ + π΄! cos ππ₯
π΅! sin ππ₯ + π΅! cos ππ₯
While we are provided this table in the reference handbook, it is often not clear what the particular form of the solution should be. Here are a few guidelines that are not in the NCEES REFERENCE HANDBOOK, but may guide you when working problems involving the METHOD OF UNDETERMINED COEFFICIENTS: β’ If f x is constant, then y! = Cx β’ If f x is linear, then y! = B! x + B! β’ If f x is quadratic, then y! = Ax ! + Bx + C β’ If f x features e!" , then y! = Be!" β’ If f x features sin cx or cos cx , then y! = B! sin cx + B! cos(cx) Note that if the driving function π π₯ , or π π₯ in our teaching approach, is π !" and the term π !" also appears in the HOMOGENEOUS SOLUTION, the PARTICULAR SOLUTION will take the form π΄π₯π !" . The take away from the table is to match the DRIVING FUNCTION in the left column with the PARTICULAR SOLUTION function in the right column.
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To find the values of the UNDETERMINED COEFFICIENTS, substitute the assumed form of π¦! (π₯) and itβs associated DERIVATIVES in to the differential equation and equate the UNKNOWN COEFFICIENTS based on the values that correspond to those terms on the opposite side of the equation. The GENERAL FORM of the COMPLETE SOLUTION of a DIFFERENTIAL EQUATION can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. It is important to note once again that the GENERAL SOLUTION for a NONHOMOGENEOUS DIFFERENTIAL EQUATION will always be made up these two unique components, the PARTICULAR SOLUTION, which may also be referred to as the NONHOMOGENEOUS SOLUTION, and the HOMEGENEOUS SOLUTION. Once they are both defined and placed together, they will represent the COMPLETE SOLUTION. The GENERAL SOLUTION, otherwise referred to as the COMPLETE SOLUTION, of a DIFFERENTIAL EQUATION is expressed as: π¦ π₯ = π¦! π₯ + π¦! (π₯) Where: β’ π¦! (π₯) is the PARTICULAR SOLUTION with π π₯ present β’ π¦! (π₯) is the HOMOGENEOUS SOLUTION corresponding to π π₯ = 0
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Once this COMPLETE SOLUTION is defined, INITIAL CONDITIONS can be used to evaluate any further unknown coefficients that remain from the HOMOEGENOUS portion of this solution.
FIRST-ORDER INSTRUMENTS: The GENERAL FORMULA for the DIFFERENTIAL EQUATION of FIRST-ORDER INSTRUMENTS can be referenced under the topic of DIFFERENTIAL EQUATIONS on page 31 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A FIRST ORDER LINEAR INSTRUMENT has an OUTPUT which is given by a NONHOMOGENEOUS FIRST ORDER LINEAR DIFFERENTIAL EQUATION, which is generally written as:
π
ππ¦ + π¦ = πΎπ₯ π‘ ππ‘
Where: β’ π is the time constant of the instrument, which is a measure of the time delay in the response to changes of input. The time constant is given in units of time β’ π¦(π‘) is the response of the system output to the forcing function β’ π₯(π‘) is the forcing function β’ πΎ is the gain of the system
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The INITIAL CONDITION for this first order system is given as: π¦ 0 = πΎπ΄ Thermometers measuring temperature are first order instruments. The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured. A first order system subjected to a constant force applied instantaneously at the initial time π‘ = 0 is represented by the step function:
π₯ π‘ =
π΄ π‘
