Question 1

One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. $y ^ { \prime } = y ^ { 2 } - 2 y - 8 ; y ( 0 ) = - 3 ; y ( 0 ) = 3$ Do these graphs represent the situation?

Accepted Answer

A) True 
 B)False

Question 2

Given the differential equation with the given initial condition: $\mathrm { y } ^ { \prime } = \mathrm { t } \cos \mathrm { t } ; \mathrm { y } ( 0 ) = 0$ is this the solution $y = t \sin t + \cos t - 1 ?$

Accepted Answer

A) True 
 B)False

Question 3

v
-An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Accepted Answer

A)   A = 60,000  e0.065te ^ { 0.065 t }e0.065t  - 52,000= $31,041.84
B)   A = 52,000  e0.065te ^ { 0.065 t }e0.065t  - 44,000= $27,969.59
C)   A = 48,000  e0.065te ^ { 0.065 t }e0.065t  - 40,000= $26,433.47
D)   A = 44,000  e0.065te ^ { 0.065 t }e0.065t  - 36,000= $24,897.34 E) A) and D)
F) A) and C)

Question 4

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $y ^ { \prime } \tan t + y \sec ^ { 2 } t = 1$ .
Is this derivative correct: $\frac { \mathrm { d } } { \mathrm { dt } } [ \mathrm { y } \tan \mathrm { t } ] = 1$ ?

Accepted Answer

A) True 
 B)False

Question 5

Which of the following functions solves the differential equation:  y′=e−2x+3?y ^ { \prime } = e ^ { - 2 x } + 3 ?y′=e−2x+3?

Accepted Answer

A)   y =  e−2xe ^ { - 2 x }e−2x  + 3
B)   y =  12\frac { 1 } { 2 }21​   e−2xe ^ { - 2 x }e−2x  + 3
C)   y = -  12\frac { 1 } { 2 }21​   e−2xe ^ { - 2 x }e−2x  + 3x
D)   none of these E) B) and C)
F) A) and C)

Question 6

Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

Accepted Answer

A) True 
 B)False

Question 7

Solve the initial value problem using an integrating factor. 
-t  y′y ^ { \prime }y′  + 3y = 5t;  y(1) =1y ( 1 )  = 1y(1) =1  , t > 0

Accepted Answer

A)   y =  54\frac { 5 } { 4 }45​  t -  14t3\frac { 1 } { 4 t ^ { 3 } }4t31​ 
B)   y =  43t2\sqrt { \frac { 4 } { 3 } t ^ { 2 } }34​t2l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z'/>​  + 1 -  43\sqrt { \frac { 4 } { 3 } }34​l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z'/>​ 
C)   y = 5  t2t ^ { 2 }t2  - 4t
D)   y = 5  t3t ^ { 3 }t3  - 4  t2t ^ { 2 }t2  E) All of the above
F) None of the above

Question 8

Suppose that a substance A is converted to substance B at a rate that is proportional to the cube of the amount of B present. The amount of A and B together is always constant, say M. If f( t)  = y is the amount of A present at time t, then which of the following differential equation describes the situation?

Accepted Answer

A)    y′y ^ { \prime }y′  =  k(M−y) 3k ( M - y )  ^ { 3 }k(M−y) 3  ; k 
B)    y′y ^ { \prime }y′  =  k(M−y) 3k ( M - y )  ^ { 3 }k(M−y) 3  ; k > 0
C)    y′y ^ { \prime }y′  =  ky3\mathrm { ky } ^ { 3 }ky3  ; k > 0
D)    y′y ^ { \prime }y′  =  ky3\mathrm { ky } ^ { 3 }ky3  ; k 
E)   none of these F) C) and E)
G) A) and B)

Question 9

A certain drug is introduced into a person's bloodstream. Suppose that the rate of decrease of the concentration of the drug in the blood is directly proportional to the product of two quantities: (a) the amount of time elapsed since the drug was introduced, and (b) the square of the concentration. Let y = f(t) denote the concentration of the drug in the blood at time t. Set up a differential equation satisfied by f(t). Does the following accurately describe this situation: $y ^ { \prime } = k t y ^ { 2 } , \text { where } k \text { is a negative constant }$

Accepted Answer

A) True 
 B)False

Question 10

Combine the terms y and $y ^ { \prime }$ into the derivative of a product, then solve the equation. $\mathrm { e } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } ^ { \prime } + 6 \mathrm { te } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } = \frac { 5 \sqrt { \mathrm { t } } } { 4 }$. Is this the solution: $y = \frac { 5 } { 6 } t ^ { 3 / 2 } e ^ { - 3 t ^ { 2 } } + C e ^ { - 3 t ^ { 2 } } ?$

Accepted Answer

A) True 
 B)False

Question 11

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below: 
Indicate whether the following statements are true or false.
-If the initial value of y(0) is 2, then the corresponding solution has an inflection point.

Accepted Answer

A) True 
 B)False

Question 12

Let f(t)  be the solution of  y′y ^ { \prime }y′  =  y2y ^ { 2 }y2  t + y +  et\mathrm { e } ^ { \mathrm { t } }et  , f(0)  = 2. If Euler's method with n = 4 is used to approximate f(t)  for  0≤t≤20 \leq t \leq 20≤t≤2  find f  (12) \left( \frac { 1 } { 2 } \right) (21​)   .

Accepted Answer

A)   2(4 +  e\sqrt { \mathrm { e } }ec-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​  ) 
B)    y22\frac { y ^ { 2 } } { 2 }2y2​  + y +  e1/2e ^ { 1 / 2 }e1/2 
C)    72\frac { 7 } { 2 }27​ 
D)   3 +  e2e ^ { 2 }e2 
E)   none of these F) None of the above
G) A) and C)

Question 13

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below: 
Indicate whether the following statements are true or false.
-y = -3, y = 1, and y = 5 are the constant solutions to $y ^ { \prime }$ = g(y).

Accepted Answer

A) True 
 B)False

Question 14

Which of the following is a sketch of the solution of  y′y ^ { \prime }y′  =  y2y ^ { 2 }y2  - 9; y(0)  = 2 ?

Accepted Answer

A)   
B)   
C)   
D)    E) None of the above
F) A) and D)

Question 15

Consider the differential equation y' = y -  y2y ^ { 2 }y2  . Which of the following statements is/are true?

Accepted Answer

A)   The function f(t)  =  1(1+e−t) \frac { 1 } { \left( 1 + e ^ { - t } \right)  }(1+e−t) 1​  is a solution to this differential equation with initial condition  y(0) =12y ( 0 )  = \frac { 1 } { 2 }y(0) =21​ 
B)   This differential equation has infinitely many solutions.
C)   The constant function f(t)  = 1 is a solution to this differential equation.
D)   If f(t)  is a solution to the differential equation satisfying the initial condition y(0)  = 0, then  f′(0) =0f ^ { \prime } ( 0 )  = 0f′(0) =0 
E)   All of these statements are true. F) C) and D)
G) B) and C)

Question 16

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition.  y′y ^ { \prime }y′  - 4y = -2  e2te ^ { 2 t }e2t  ; y(0)  = -1

Accepted Answer

A)   integrating factor:  e−4te ^ { - 4 t }e−4t General solution:  y=−e−2t+Ce4ty = - e ^ { - 2 t } + C ^ { e ^ { 4 t } }y=−e−2t+Ce4t Particular solution: y = -  e−2te ^ { - 2 t }e−2t  - 2  e4te ^ { 4 t }e4t 
B)   integrating factor:  e−4te ^ { - 4 t }e−4t General solution: y =  e2te ^ { 2 t }e2t  + C  e4te ^ { 4 t }e4t Particular solution: y =  e2te ^ { 2 t }e2t  - 2  e4te ^ { 4 t }e4t 
C)   integrating factor:  e4te ^ { 4 t }e4t General solution: y = -2t + C  e4te ^ { 4 t }e4t Particular solution: y = -2t -  e4te ^ { 4 t }e4t 
D)   integrating factor:  e4te ^ { 4 t }e4t General solution: y =  13\frac { 1 } { 3 }31​   e6te ^ { 6 t }e6t  + C  e4te ^ { 4 t }e4t Particular solution: y =  13\frac { 1 } { 3 }31​   e6te ^ { 6 t }e6t  -  43\frac { 4 } { 3 }34​   e4te ^ { 4 t }e4t Solve the equation using an integrating factor. E) A) and D)
F) A) and C)

Question 17

Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1).
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Use Euler's method with n = 2...

Question 18

A savings account earns 6% annual interest, compounded continuously. An initial deposit of $8500 is made, and thereafter money is withdrawn continuously at the rate of $480 per year. Does the following accurately represent this situation: $y ^ { \prime } = 0.06 y - 480 ; y ( 0 ) = 8500 ?$

Accepted Answer

A) True 
 B)False

Question 19

Solve the differential equation with the given initial condition. 
- y′=3t2(4−y) 2,y(0) =2y ^ { \prime } = 3 t ^ { 2 } ( 4 - y )  ^ { 2 } , y ( 0 )  = 2y′=3t2(4−y) 2,y(0) =2

Accepted Answer

A)   y = 4 -  1t3+12\frac { 1 } { t ^ { 3 } + \frac { 1 } { 2 } }t3+21​1​ 
B)   y =  1t3+2\frac { 1 } { t ^ { 3 } + 2 }t3+21​ 
C)   y =  t3t ^ { 3 }t3  +  12\frac { 1 } { 2 }21​ 
D)   y = 4 +  t3t ^ { 3 }t3E) A) and B)
F) None of the above

Question 20

Solve the differential equation with the given initial condition. 
- y′=tan⁡tsec⁡2t;y(0) =1y ^ { \prime } = 	an t \sec ^ { 2 } t ; y ( 0 )  = 1y′=tantsec2t;y(0) =1

Accepted Answer

A)   y = ln  ∣tan⁡t∣| 	an \mathrm { t } |∣tant∣  + 1
B)   y =  tan⁡2t2\frac { 	an ^ { 2 } \mathrm { t } } { 2 }2tan2t​  + 1
C)   y = tan t + 1
D)   y =  sec⁡2t3\frac { \sec ^ { 2 } t } { 3 }3sec2t​  +  23\frac { 2 } { 3 }32​E) C) and D)
F) All of the above

Exam 10: Differential Equations

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $y ^ { \prime } \tan t + y \sec ^ { 2 } t = 1$ . Is this derivative correct: $\frac { \mathrm { d } } { \mathrm { dt } } [ \mathrm { y } \tan \mathrm { t } ] = 1$ ?

Correct Answer
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Given the differential equation with the given initial condition: $\mathrm { y } ^ { \prime } = \mathrm { t } \cos \mathrm { t } ; \mathrm { y } ( 0 ) = 0$ is this the solution $y = t \sin t + \cos t - 1 ?$

Correct Answer
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Let f(t) be the solution of $y ^ { \prime }$ = $y ^ { 2 }$ t + y + $\mathrm { e } ^ { \mathrm { t } }$ , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for $0 \leq t \leq 2$ find f $\left( \frac { 1 } { 2 } \right)$ .

Correct Answer
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Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

Correct Answer
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Solve the differential equation with the given initial condition. - $y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2$

Correct Answer
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Which of the following is a sketch of the solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 9; y(0) = 2 ?

Correct Answer
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Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

Correct Answer
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Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. $y ^ { \prime }$ - 4y = -2 $e ^ { 2 t }$ ; y(0) = -1

Correct Answer
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Solve the initial value problem using an integrating factor. -t $y ^ { \prime }$ + 3y = 5t; $y ( 1 ) = 1$ , t > 0

Correct Answer
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Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1). Enter just a reduced fraction of form $\frac { a } { b }$ .

Correct Answer
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v -An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Correct Answer
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A savings account earns 6% annual interest, compounded continuously. An initial deposit of $8500 is made, and thereafter money is withdrawn continuously at the rate of $480 per year. Does the following accurately represent this situation: $y ^ { \prime } = 0.06 y - 480 ; y ( 0 ) = 8500 ?$

Correct Answer
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Which of the following functions solves the differential equation: $y ^ { \prime } = e ^ { - 2 x } + 3 ?$

Correct Answer
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Solve the differential equation with the given initial condition. - $y ^ { \prime } = \tan t \sec ^ { 2 } t ; y ( 0 ) = 1$

Correct Answer
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Exam 10: Differential Equations

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Given the differential equation with the given initial condition: y′=tcos⁡t;y(0)=0\mathrm { y } ^ { \prime } = \mathrm { t } \cos \mathrm { t } ; \mathrm { y } ( 0 ) = 0y′=tcost;y(0)=0 is this the solution y=tsin⁡t+cos⁡t−1?y = t \sin t + \cos t - 1 ?y=tsint+cost−1?

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Let f(t) be the solution of y′y ^ { \prime }y′ = y2y ^ { 2 }y2 t + y + et\mathrm { e } ^ { \mathrm { t } }et , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for 0≤t≤20 \leq t \leq 20≤t≤2 find f (12) \left( \frac { 1 } { 2 } \right) (21​) .

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Consider the differential equation y' = y - y2y ^ { 2 }y2 . Which of the following statements is/are true?

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Solve the differential equation with the given initial condition. - y′=3t2(4−y) 2,y(0) =2y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2y′=3t2(4−y) 2,y(0) =2

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Which of the following is a sketch of the solution of y′y ^ { \prime }y′ = y2y ^ { 2 }y2 - 9; y(0) = 2 ?

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Given the differential equation: ty′=ln⁡tty ^ { \prime } = \ln tty′=lnt , is this the solution y=(ln⁡t)22+C?y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?y=2(lnt)2​+C?

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Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. y′y ^ { \prime }y′ - 4y = -2 e2te ^ { 2 t }e2t ; y(0) = -1

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Solve the initial value problem using an integrating factor. -t y′y ^ { \prime }y′ + 3y = 5t; y(1) =1y ( 1 ) = 1y(1) =1 , t > 0

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Correct AnswerverifiedShow Answer

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Consider the differential equation y′y ^ { \prime }y′ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -If the initial value of y(0) is 2, then the corresponding solution has an inflection point.

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Use Euler's method with n = 2 to approximate the solution f(t) to y′=2y−t,y(0)=1y ^ { \prime } = 2 y - t , y ( 0 ) = 1y′=2y−t,y(0)=1 Estimate f(1). Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

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Correct AnswerverifiedShow Answer

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Consider the differential equation y′y ^ { \prime }y′ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -y = -3, y = 1, and y = 5 are the constant solutions to y′y ^ { \prime }y′ = g(y).

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v -An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Which of the following functions solves the differential equation: y′=e−2x+3?y ^ { \prime } = e ^ { - 2 x } + 3 ?y′=e−2x+3?

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Solve the differential equation with the given initial condition. - y′=tan⁡tsec⁡2t;y(0) =1y ^ { \prime } = \tan t \sec ^ { 2 } t ; y ( 0 ) = 1y′=tantsec2t;y(0) =1

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Correct Answer
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Given the differential equation with the given initial condition: $\mathrm { y } ^ { \prime } = \mathrm { t } \cos \mathrm { t } ; \mathrm { y } ( 0 ) = 0$ is this the solution $y = t \sin t + \cos t - 1 ?$

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Let f(t) be the solution of $y ^ { \prime }$ = $y ^ { 2 }$ t + y + $\mathrm { e } ^ { \mathrm { t } }$ , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for $0 \leq t \leq 2$ find f $\left( \frac { 1 } { 2 } \right)$ .

Correct Answer
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Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

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Solve the differential equation with the given initial condition. - $y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2$

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Which of the following is a sketch of the solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 9; y(0) = 2 ?

Correct Answer
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Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

Correct Answer
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Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. $y ^ { \prime }$ - 4y = -2 $e ^ { 2 t }$ ; y(0) = -1

Correct Answer
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Solve the initial value problem using an integrating factor. -t $y ^ { \prime }$ + 3y = 5t; $y ( 1 ) = 1$ , t > 0

Correct Answer
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Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1). Enter just a reduced fraction of form $\frac { a } { b }$ .

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Which of the following functions solves the differential equation: $y ^ { \prime } = e ^ { - 2 x } + 3 ?$

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Solve the differential equation with the given initial condition. - $y ^ { \prime } = \tan t \sec ^ { 2 } t ; y ( 0 ) = 1$

Correct Answer
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