66.1 Separable Differential Equations Concept Overview
SEPARABLE DIFFERENTIAL EQUATIONS | CONCEPT OVERVIEW The TOPIC of SEPARABLE DIFFERENTIAL EQUATIONS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
CONCEPT INTRO: A SEPARABLE DIFFERENTIAL EQUATION is a differential equation that can be written to isolate the INDEPENDENT and DEPENDENT VARIABLES together with their differentials on opposites sides of the equation, such that: π₯ ππ₯ = π¦ ππ¦ The driving characteristic of what makes these DIFFERENTIAL EQUATIONS fall under the category of being SEPARABLE is this fact that the INDEPENDENT and DEPENDENT VARIABLES can be moved to their own dedicated side within the equation. The STANDARD FORM of a DIFFERENTIAL EUQATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
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A SEPARABLE DIFFERENTIAL EQUATION may also expressed in any one of the following forms, all of which are equivalent: π π₯ ππ₯ + π π¦ ππ¦ = 0 π π¦ ππ¦ = βπ π₯ ππ₯
π π¦
ππ¦ = βπ(π₯) ππ₯
β« π π₯ ππ₯ + β« π π¦ ππ¦ = πΆ Where: β’ π(π₯) is a function of βπ₯β only or a constant β’ π(π¦) is a function of βπ¦β only or a constant β’ πΆ represents an arbitrary constant When working with a SEPARABLE DIFFERENTIAL EQUATION, we will pursue either an IMPLICIT or EXPLICIT SOLUTION. An IMPLICIT SOLUTION is one which will result in an expression where the INDEPENDENT and DEPENDENT VARIABLES are not isolated in their pure form on opposite sides of the equation.
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We can think of an IMPLICIT SOLUTION as a solution where there will be some combination of both the INDEPENDENT and DEPENDENT VARIABLES on one or both sides of the solutionβs expression. In other cases, the DEPENDENT VARIABLE may be expressed as a function in and of itself, and not just strictly a function of the INDEPENDENT VARIABLE. An EXPLICIT SOLUTION, on the other hand, is one which will result in an expression where the INDEPENDENT and DEPENDENT VARIABLES can be isolated in their pure form on opposite sides of the equation. We can think of an EXPLICIT SOLUTION as a solution where each side of the equation will have only INDEPENDENT or only DEPENDENT VARIABLESβ¦they will be fully isolated from on another. The goal of SEPARABLE DIFFERENTIAL EQUATIONS is the ability to integrate the equation with respect to βπ¦β on one side while integrating the equation with respect to βπ₯β on the other. The process will generally be the same for every SEPERABLE DIFFERENTIAL EQUATION that we encounter, and can be roughly stated as: 1. Rewrite the differential equation such that all functions and differentials of the independent variable, βπ₯β, are one side of the equation, and all functions and differentials of the dependent variable, βπ¦" are on the opposite side of the equation.
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For example, given the function: π π₯ ππ₯ + π π¦ ππ¦ = 0 We will rewrite it as: π π¦ ππ¦ = π π₯ ππ₯ 2. Integrate both sides of the equation with respect to their differential, such that: β« π π¦ ππ¦ = β« π π₯ ππ₯ + πΆ After integration, we find the resulting general solution is the IMPLICIT SOLUTION of the equation, where βπ¦β is defined implicitly as a function of βπ₯β by the equation relating the integrals, with respect to their individual variables, of π(π₯) and π(π¦). 3. Define the EXPLICIT SOLUTION where possible, and as requested, by organizing the expression so that the INDEPENDENT and DEPENDENT VARIABLES are isolated on either side of the equation. This expression will take the strict form of π¦ = π(π₯), where βπ¦β is EXPLICITLY defined by a function π π₯ . 4. Determine the PARTICULAR SOLUTION as necessary, using INITIAL VALUES or other defined BOUNDARY CONDITIONS.
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SEPARABLE DIFFERENTIAL EQUATIONS | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The explicit particular solution of the following differential equation is best represented as:
π!
ππ¦ β π₯ β π₯ ! = 0; π¦ 1 = 2 ππ₯
A. π¦ π₯ = ln B. π¦ π₯ = ln C. π¦ π₯ = ln D. π¦ π₯ = ln
!! ! !! ! !! ! !! !
+ + + +
!! ! !! ! !! ! !! !
!
+ π! β ! !
+ π! β ! + π! !
+ π! β !
SOLUTION: The TOPIC of SEPARABLE DIFFERENTIAL EQUATIONS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
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by Prepineer | Prepineer.com
A SEPARABLE DIFFERENTIAL EQUATION is a differential equation that can be written to isolate the INDEPENDENT and DEPENDENT VARIABLES together with their differentials on opposites sides of the equation, such that: π₯ ππ₯ = π¦ ππ¦ The driving characteristic of what makes these DIFFERENTIAL EQUATIONS fall under the category of being SEPARABLE is this fact that the INDEPENDENT and DEPENDENT VARIABLES can be moved to their dedicated side within the equation. In this problem we are given the DIFFERENTIAL EQUATION:
π!
ππ¦ β π₯ β π₯! = 0 ππ₯
At first glance we can confirm that this differential equation is in fact SEPARABLE, and can be separated such that all the x-terms and all the y-terms are isolated on their dedicated side of the equation Rearranging the DIFFERENTIAL EQUATION to fall in line with this characteristic of a SEPERABLE DIFFERENTIAL EQUATION, we can tweak the x and y terms such that we have: π ! ππ¦ = π₯ ! + π₯ ππ₯
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Our next step in this problem is to integrate both sides with respect to the variable and differential that is represented: β« π ! ππ¦ = β« π₯ ! + π₯ ππ₯ Evaluating the integral, we find that the IMPLICIT SOLUTION of the SEPERABLE DIFFERENTIAL EQUATION can be written as: π₯! π₯! π = + +πΆ 4 2 !
We can now determine the EXPLICIT SOLUTION. Recall that the EXPLICIT SOLUTION organizes the expression so that the INDEPENDENT and DEPENDENT VARIABLES are isolated on either side of the equation and takes the strict form of π¦ = π(π₯), where π¦ is explicitly defined by a function π π₯ . We are close here, but do not have an expression that is EXPLICIT due to the exponent of y being on the left side of the equationβ¦we have to tackle that. Doing so, we can rewrite the formula as: π₯! π₯! π¦(π₯) = ln + +πΆ 4 2
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With the EXPLICIT form of the general solution defined, we can now employ the provided INITIAL CONDITION solve for the PARTICULAR SOLUTION. We are given the INITIAL CONDITION: π¦ (1) = 2 Plugging this in to our EXPLICIT SOLUTION, we get: 1! 1! 2 = ln + +πΆ 4 2 Rearranging to solve for the CONSTANT OF INTEGRATION, we find that:
πΆ = π! β
3 4
Inserting this CONSTANT OF INTEGRATION into the EXPLICIT FORM of the general solution, we get: π₯! π₯! 3 π¦ π₯ = ln + + π! β 4 2 4
The correct answer choice is B. π² π± = π₯π§
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π±π π
+
π±π π
π
+ ππ β π
by Prepineer | Prepineer.com
CONCEPT INTRO: A SEPARABLE DIFFERENTIAL EQUATION is a differential equation that can be written to isolate the INDEPENDENT and DEPENDENT VARIABLES together with their differentials on opposites sides of the equation, such that: π₯ ππ₯ = π¦ ππ¦ The driving characteristic of what makes these DIFFERENTIAL EQUATIONS fall under the category of being SEPARABLE is this fact that the INDEPENDENT and DEPENDENT VARIABLES can be moved to their own dedicated side within the equation. The STANDARD FORM of a DIFFERENTIAL EUQATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
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by Prepineer | Prepineer.com
A SEPARABLE DIFFERENTIAL EQUATION may also expressed in any one of the following forms, all of which are equivalent: π π₯ ππ₯ + π π¦ ππ¦ = 0 π π¦ ππ¦ = βπ π₯ ππ₯
π π¦
ππ¦ = βπ(π₯) ππ₯
β« π π₯ ππ₯ + β« π π¦ ππ¦ = πΆ Where: β’ π(π₯) is a function of βπ₯β only or a constant β’ π(π¦) is a function of βπ¦β only or a constant β’ πΆ represents an arbitrary constant When working with a SEPARABLE DIFFERENTIAL EQUATION, we will pursue either an IMPLICIT or EXPLICIT SOLUTION. An IMPLICIT SOLUTION is one which will result in an expression where the INDEPENDENT and DEPENDENT VARIABLES are not isolated in their pure form on opposite sides of the equation.
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by Prepineer | Prepineer.com
We can think of an IMPLICIT SOLUTION as a solution where there will be some combination of both the INDEPENDENT and DEPENDENT VARIABLES on one or both sides of the solutionβs expression. In other cases, the DEPENDENT VARIABLE may be expressed as a function in and of itself, and not just strictly a function of the INDEPENDENT VARIABLE. An EXPLICIT SOLUTION, on the other hand, is one which will result in an expression where the INDEPENDENT and DEPENDENT VARIABLES can be isolated in their pure form on opposite sides of the equation. We can think of an EXPLICIT SOLUTION as a solution where each side of the equation will have only INDEPENDENT or only DEPENDENT VARIABLESβ¦they will be fully isolated from on another. The goal of SEPARABLE DIFFERENTIAL EQUATIONS is the ability to integrate the equation with respect to βπ¦β on one side while integrating the equation with respect to βπ₯β on the other. The process will generally be the same for every SEPERABLE DIFFERENTIAL EQUATION that we encounter, and can be roughly stated as: 1. Rewrite the differential equation such that all functions and differentials of the independent variable, βπ₯β, are one side of the equation, and all functions and differentials of the dependent variable, βπ¦" are on the opposite side of the equation.
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by Prepineer | Prepineer.com
For example, given the function: π π₯ ππ₯ + π π¦ ππ¦ = 0 We will rewrite it as: π π¦ ππ¦ = π π₯ ππ₯ 2. Integrate both sides of the equation with respect to their differential, such that: β« π π¦ ππ¦ = β« π π₯ ππ₯ + πΆ After integration, we find the resulting general solution is the IMPLICIT SOLUTION of the equation, where βπ¦β is defined implicitly as a function of βπ₯β by the equation relating the integrals, with respect to their individual variables, of π(π₯) and π(π¦). 3. Define the EXPLICIT SOLUTION where possible, and as requested, by organizing the expression so that the INDEPENDENT and DEPENDENT VARIABLES are isolated on either side of the equation. This expression will take the strict form of π¦ = π(π₯), where βπ¦β is EXPLICITLY defined by a function π π₯ . 4. Determine the PARTICULAR SOLUTION as necessary, using INITIAL VALUES or other defined BOUNDARY CONDITIONS.
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SEPARABLE DIFFERENTIAL EQUATIONS | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The explicit particular solution of the following differential equation is best represented as:
π!
ππ¦ β π₯ β π₯ ! = 0; π¦ 1 = 2 ππ₯
A. π¦ π₯ = ln B. π¦ π₯ = ln C. π¦ π₯ = ln D. π¦ π₯ = ln
!! ! !! ! !! ! !! !
+ + + +
!! ! !! ! !! ! !! !
!
+ π! β ! !
+ π! β ! + π! !
+ π! β !
SOLUTION: The TOPIC of SEPARABLE DIFFERENTIAL EQUATIONS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
Made with
by Prepineer | Prepineer.com
A SEPARABLE DIFFERENTIAL EQUATION is a differential equation that can be written to isolate the INDEPENDENT and DEPENDENT VARIABLES together with their differentials on opposites sides of the equation, such that: π₯ ππ₯ = π¦ ππ¦ The driving characteristic of what makes these DIFFERENTIAL EQUATIONS fall under the category of being SEPARABLE is this fact that the INDEPENDENT and DEPENDENT VARIABLES can be moved to their dedicated side within the equation. In this problem we are given the DIFFERENTIAL EQUATION:
π!
ππ¦ β π₯ β π₯! = 0 ππ₯
At first glance we can confirm that this differential equation is in fact SEPARABLE, and can be separated such that all the x-terms and all the y-terms are isolated on their dedicated side of the equation Rearranging the DIFFERENTIAL EQUATION to fall in line with this characteristic of a SEPERABLE DIFFERENTIAL EQUATION, we can tweak the x and y terms such that we have: π ! ππ¦ = π₯ ! + π₯ ππ₯
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Our next step in this problem is to integrate both sides with respect to the variable and differential that is represented: β« π ! ππ¦ = β« π₯ ! + π₯ ππ₯ Evaluating the integral, we find that the IMPLICIT SOLUTION of the SEPERABLE DIFFERENTIAL EQUATION can be written as: π₯! π₯! π = + +πΆ 4 2 !
We can now determine the EXPLICIT SOLUTION. Recall that the EXPLICIT SOLUTION organizes the expression so that the INDEPENDENT and DEPENDENT VARIABLES are isolated on either side of the equation and takes the strict form of π¦ = π(π₯), where π¦ is explicitly defined by a function π π₯ . We are close here, but do not have an expression that is EXPLICIT due to the exponent of y being on the left side of the equationβ¦we have to tackle that. Doing so, we can rewrite the formula as: π₯! π₯! π¦(π₯) = ln + +πΆ 4 2
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With the EXPLICIT form of the general solution defined, we can now employ the provided INITIAL CONDITION solve for the PARTICULAR SOLUTION. We are given the INITIAL CONDITION: π¦ (1) = 2 Plugging this in to our EXPLICIT SOLUTION, we get: 1! 1! 2 = ln + +πΆ 4 2 Rearranging to solve for the CONSTANT OF INTEGRATION, we find that:
πΆ = π! β
3 4
Inserting this CONSTANT OF INTEGRATION into the EXPLICIT FORM of the general solution, we get: π₯! π₯! 3 π¦ π₯ = ln + + π! β 4 2 4
The correct answer choice is B. π² π± = π₯π§
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π±π π
+
π±π π
π
+ ππ β π
by Prepineer | Prepineer.com
