Question 1

Show that the curves are orthogonal.
-$y ^ { 2 } + x ^ { 2 } = 1 \text { and } x y = 2$

Accepted Answer

The answer of Show that the curves are orthogonal.
-\(y ^...

Question 2

Obtain a slope field and add to its graphs of the solution curves passing through the given points.
-$$y ^ { \prime } = - y \text { with } ( 0,2 )$$

Accepted Answer

The answer of Obtain a slope field and add to...

Question 3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method with the specified step size to estimate the value of the solution at the given point $x$. Find the value of the exact solution at $x ^ { * }$.
-$$\mathrm { y } ^ { \prime } = 4 \mathrm { xe } ^ { \mathrm { x } ^ { 4 } } , \mathrm { y } ( 0 ) = 4 , \mathrm { dx } = 0.1 , \mathrm { x } ^ { * } = 1$$

Accepted Answer

A)  Euler's method gives $\mathrm { y } \approx 5.7183$; the exact solution is $6.2619$ 
B)  Euler's method gives $\mathrm { y } \approx 7.0655$; the exact solution is $5.9901$ 
C)  Euler's method gives $\mathrm { y } \approx 6.4232$; the exact solution is $5.7183$ 
D)  Euler's method gives $\mathrm { y } \approx 5.7808$; the exact solution is $5.4465$ 
A)  Euler's method gives $\mathrm { y } \approx 5.7183$; the exact solution is $6.2619$ 
B)  Euler's method gives $\mathrm { y } \approx 7.0655$; the exact solution is $5.9901$ 
C)  Euler's method gives $\mathrm { y } \approx 6.4232$; the exact solution is $5.7183$ 
D)  Euler's method gives $\mathrm { y } \approx 5.7808$; the exact solution is $5.4465$

Question 4

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places.
-$$y ^ { \prime } = 1 + \frac { y } { x } , y ( 2 ) = - 3 , d x = 0.5$$

Accepted Answer

A)  y1 = -4.8750, y2 = -4.0800, y3 = -4.1600 
B)  y1 = -3.2500, y2 = -3.4000, y3 = -3.4667 
C)  y1 = -6.5000, y2 = -6.8000, y3 = -13.8667 
D)  y1 = -6.5000, y2 = -5.1000, y3 = -6.9333 
A)  y1 = -4.8750, y2 = -4.0800, y3 = -4.1600 
B)  y1 = -3.2500, y2 = -3.4000, y3 = -3.4667 
C)  y1 = -6.5000, y2 = -6.8000, y3 = -13.8667 
D)  y1 = -6.5000, y2 = -5.1000, y3 = -6.9333

Question 5

Identify equilibrium values and determine which are stable and which are unstable.
-$$y ^ { \prime } = \sqrt { 7 y } , y > 0$$

Accepted Answer

A)  y = 0 is an unstable equilibrium value. 
B)  y = 7 is an unstable equilibrium value. 
C)  y = 0 is a stable equilibrium value. 
D)  There are no equilibrium values. 
A)  y = 0 is an unstable equilibrium value. 
B)  y = 7 is an unstable equilibrium value. 
C)  y = 0 is a stable equilibrium value. 
D)  There are no equilibrium values.

Question 6

Provide an appropriate response.
-A catamaran is running along a course with the wind providing a constant force of 30 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to Two times the boat's speed, and the initial velocity is 1 ft/sec. What is the maximum velocity in feet per second Of the boat under this wind?

Accepted Answer

A)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 0$ or $\mathrm { v } = 12 \mathrm { ft } / \mathrm { sec }$ 
B)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 0$ or $\mathrm { v } = 15 \mathrm { ft } / \mathrm { sec }$ 
C)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 15$ or $\mathrm { v } = 0 \mathrm { ft } / \mathrm { sec }$ 
D)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 12$ or $\mathrm { v } = 0 \mathrm { ft } / \mathrm { sec }$ 
A)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 0$ or $\mathrm { v } = 12 \mathrm { ft } / \mathrm { sec }$ 
B)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 0$ or $\mathrm { v } = 15 \mathrm { ft } / \mathrm { sec }$ 
C)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 15$ or $\mathrm { v } = 0 \mathrm { ft } / \mathrm { sec }$ 
D)  The maximum velocity occurs when $\frac { \mathrm { dv } } { \mathrm { dt } } = 12$ or $\mathrm { v } = 0 \mathrm { ft } / \mathrm { sec }$

Question 7

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places.
-$$y ^ { \prime } = y ^ { 2 } ( 1 - 2 x ) , y ( - 1 ) = - 1 , d x = 0.5$$

Accepted Answer

A)  y1 = 0.4, y2 = 0.63, y3 = 0.7472 
B)  y1 = 0.7, y2 = 0.99, y3 = 1.9136 
C)  y1 = 0.5, y2 = 0.75, y3 = 1.03125 
D)  y1 = 0.7, y2 = 0.99, y3 = 1.2656 
A)  y1 = 0.4, y2 = 0.63, y3 = 0.7472 
B)  y1 = 0.7, y2 = 0.99, y3 = 1.9136 
C)  y1 = 0.5, y2 = 0.75, y3 = 1.03125 
D)  y1 = 0.7, y2 = 0.99, y3 = 1.2656

Question 8

Solve the differential equation.
-$$\cos x \frac { d y } { d x } + y \sin x = \sin x \cos x$$

Accepted Answer

Question 9

Solve the differential equation.
-$y ^ { \prime } - y = - x y ^ { 2 }$

Accepted Answer

A)  $x - 1 + C$ 
B)  $\frac { e ^ { x } } { ( x + 1 ) + C }$ 
C) $ \frac { e ^ { X } } { e ^ { X } ( x - 1 ) + C }$ 
D)  $x + 1 + C$ 
A)  $x - 1 + C$ 
B)  $\frac { e ^ { x } } { ( x + 1 ) + C }$ 
C) $ \frac { e ^ { X } } { e ^ { X } ( x - 1 ) + C }$ 
D)  $x + 1 + C$

Question 10

The autonomous differential equation represents a model for population growth. Use phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable?
-$\frac { \mathrm { dP } } { \mathrm { dt } } = 1 - 8 \mathrm { P }$

Accepted Answer

The answer of The autonomous differential equation represents a model...

Question 11

Construct a phase line. Identify signs of y and y .
-$\mathrm { y } ^ { \prime } = \sqrt { 3 \mathrm { y } } , \quad \mathrm { y } > 0$

Accepted Answer

A) 
  
B) 
  
C)

D) 
  
A) 
  
B) 
  
C)

D)

Question 12

Solve the problem.
-A tank initially contains 100 gal of brine in which 40 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and flows out of the tank at
The rate of 3 gal/min. Find the solution to the differential equation that models the mixing process.

Accepted Answer

A)  $y = 2 ( 100 + t ) - \frac { 10 ^ { 8 } } { ( 100 + t ) ^ { 3 } }$ 
B)  $y = 4 ( 50 + t ) - \frac { C } { ( 100 + t ) ^ { 3 } }$ 
C)  $y = 4 ( 50 + t ) - \frac { 10 ^ { 8 } } { ( 100 + t ) ^ { 3 } }$ 
D)  $y = 2 ( 100 + t ) - \frac { C } { \left( 100 + t ^ { 3 } \right) }$ 
A)  $y = 2 ( 100 + t ) - \frac { 10 ^ { 8 } } { ( 100 + t ) ^ { 3 } }$ 
B)  $y = 4 ( 50 + t ) - \frac { C } { ( 100 + t ) ^ { 3 } }$ 
C)  $y = 4 ( 50 + t ) - \frac { 10 ^ { 8 } } { ( 100 + t ) ^ { 3 } }$ 
D)  $y = 2 ( 100 + t ) - \frac { C } { \left( 100 + t ^ { 3 } \right) }$

Question 13

Determine which of the following equations is correct.
-$$x \int \frac { 1 } { 4 x } d x =$$

Accepted Answer

A)  $\frac { x } { 4 } + C$ 
B)  $\frac { x ^ { 2 } \ln | x | } { 8 } + C$ 
C)  $4 x \ln | x | + C$ 
D)  $\frac { x } { 4 } \ln | x | + C$ 
A)  $\frac { x } { 4 } + C$ 
B)  $\frac { x ^ { 2 } \ln | x | } { 8 } + C$ 
C)  $4 x \ln | x | + C$ 
D)  $\frac { x } { 4 } \ln | x | + C$

Question 14

Solve the differential equation.
-$$x \frac { d y } { d x } = \frac { \cos x } { x ^ { 2 } } - 3 y , x > 0$$

Accepted Answer

A)  $y = x ^ { 3 } ( \sin x + C ) , x > 0$ 
B)  $y = x ^ { - 3 } ( \sin x + C ) , x > 0$ 
C)  $y = x ^ { - 3 } ( \cos x + C ) , x > 0$ 
D)  $y = x ^ { 3 } ( \cos x + C ) , x > 0$ 
A)  $y = x ^ { 3 } ( \sin x + C ) , x > 0$ 
B)  $y = x ^ { - 3 } ( \sin x + C ) , x > 0$ 
C)  $y = x ^ { - 3 } ( \cos x + C ) , x > 0$ 
D)  $y = x ^ { 3 } ( \cos x + C ) , x > 0$

Question 15

Solve.
-A 60-kg skateboarder on a 1-kg board starts coasting on level ground at 8 m/sec. Let k = 3.2 kg/sec. About how far will the skater coast before reaching a complete stop?

Accepted Answer

A)  1536.00 m 
B)  152.50 m 
C)  24.00 m 
D)  150.00 m 
A)  1536.00 m 
B)  152.50 m 
C)  24.00 m 
D)  150.00 m

Question 16

Solve the differential equation.
-$$4 y ^ { \prime } = e ^ { x / 4 } + y$$

Accepted Answer

A)  $y = \frac { - x e ^ { x / 4 } + C e ^ { x / 4 } } { 4 }$ 
B)  $y = \frac { x e ^ { x / 4 } + C e ^ { x / 4 } } { 4 }$ 
C)  $y = \frac { x e ^ { x / 4 } + C } { 4 }$ 
D)  $y = x e ^ { x / 4 } + C e ^ { x / 4 }$ 
A)  $y = \frac { - x e ^ { x / 4 } + C e ^ { x / 4 } } { 4 }$ 
B)  $y = \frac { x e ^ { x / 4 } + C e ^ { x / 4 } } { 4 }$ 
C)  $y = \frac { x e ^ { x / 4 } + C } { 4 }$ 
D)  $y = x e ^ { x / 4 } + C e ^ { x / 4 }$

Question 17

Solve. Round your results to four decimal places.
-Use the Euler method with $d x = 0.2$ to estimate $y ( 3 )$ if $y ^ { \prime } = - y / x$ and $y ( 2 ) = 3$. What is the exact value of $y ( 3 )$ ?

Accepted Answer

A)  $\mathrm { y } \approx 0.7895$, exact value is $2.0000$ 
B)  $\mathrm { y } \approx 1.9286$, exact value is $1.5000$ 
C)  $\mathrm { y } \approx 1.1053$, exact value is $6.0000$ 
D)  $y \approx 1.7368$, exact value is $3.0000$ 
A)  $\mathrm { y } \approx 0.7895$, exact value is $2.0000$ 
B)  $\mathrm { y } \approx 1.9286$, exact value is $1.5000$ 
C)  $\mathrm { y } \approx 1.1053$, exact value is $6.0000$ 
D)  $y \approx 1.7368$, exact value is $3.0000$

Question 18

Match the differential equation with the appropriate slope field.
-$$y ^ { \prime } = ( y + 3 ) ( y - 3 )$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Obtain a slope field and add to its graphs of the solution curves passing through the given points.
-$$y ^ { \prime } = - y ^ { 2 } \text { with } ( 0,1 )$$

Accepted Answer

The answer of Obtain a slope field and add to...

Question 20

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method with the specified step size to estimate the value of the solution at the given point $x$. Find the value of the exact solution at $x ^ { * }$.
-Using the given conditions, obtain a slope field, solve for the general solution, and plot solution curves for the arbitrary constant values $C = - 2,2$, and 4 .
$$\begin{array} { l } 
y ^ { \prime } = \frac { 4 x ^ { 2 } + 3 x + 1 } { 2 ( y - 1 ) } \
- 3 \leq x \leq 3 , - 3 \leq y \leq 3
\end{array}$$

Accepted Answer

The answer of  MULTIPLE CHOICE. Choose the one alternative...

Show that the curves are orthogonal. - $y ^ { 2 } + x ^ { 2 } = 1 \text { and } x y = 2$

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = - y \text { with } ( 0,2 )$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - $y ^ { \prime } = 1 + \frac { y } { x } , y ( 2 ) = - 3 , d x = 0.5$

Identify equilibrium values and determine which are stable and which are unstable. - $y ^ { \prime } = \sqrt { 7 y } , y > 0$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - $y ^ { \prime } = y ^ { 2 } ( 1 - 2 x ) , y ( - 1 ) = - 1 , d x = 0.5$

Solve the differential equation. - $\cos x \frac { d y } { d x } + y \sin x = \sin x \cos x$

Solve the differential equation. - $y ^ { \prime } - y = - x y ^ { 2 }$

Construct a phase line. Identify signs of y and y . - $\mathrm { y } ^ { \prime } = \sqrt { 3 \mathrm { y } } , \quad \mathrm { y } > 0$

Determine which of the following equations is correct. - $x \int \frac { 1 } { 4 x } d x =$

Solve the differential equation. - $x \frac { d y } { d x } = \frac { \cos x } { x ^ { 2 } } - 3 y , x > 0$

Solve. -A 60-kg skateboarder on a 1-kg board starts coasting on level ground at 8 m/sec. Let k = 3.2 kg/sec. About how far will the skater coast before reaching a complete stop?

Solve the differential equation. - $4 y ^ { \prime } = e ^ { x / 4 } + y$

Solve. Round your results to four decimal places. -Use the Euler method with $d x = 0.2$ to estimate $y ( 3 )$ if $y ^ { \prime } = - y / x$ and $y ( 2 ) = 3$ . What is the exact value of $y ( 3 )$ ?

Match the differential equation with the appropriate slope field. - $y ^ { \prime } = ( y + 3 ) ( y - 3 )$

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = - y ^ { 2 } \text { with } ( 0,1 )$

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Exam 10: First-Order Differential Equations

Show that the curves are orthogonal. - y2+x2=1 and xy=2y ^ { 2 } + x ^ { 2 } = 1 \text { and } x y = 2y2+x2=1 and xy=2

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - y′=−y with (0,2)y ^ { \prime } = - y \text { with } ( 0,2 )y′=−y with (0,2)

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - y′=1+yx,y(2)=−3,dx=0.5y ^ { \prime } = 1 + \frac { y } { x } , y ( 2 ) = - 3 , d x = 0.5y′=1+xy​,y(2)=−3,dx=0.5

Identify equilibrium values and determine which are stable and which are unstable. - y′=7y,y>0y ^ { \prime } = \sqrt { 7 y } , y > 0y′=7y​,y>0

Solve the differential equation. - cos⁡xdydx+ysin⁡x=sin⁡xcos⁡x\cos x \frac { d y } { d x } + y \sin x = \sin x \cos xcosxdxdy​+ysinx=sinxcosx

Solve the differential equation. - y′−y=−xy2y ^ { \prime } - y = - x y ^ { 2 }y′−y=−xy2

Construct a phase line. Identify signs of y and y . - y′=3y,y>0\mathrm { y } ^ { \prime } = \sqrt { 3 \mathrm { y } } , \quad \mathrm { y } > 0y′=3y​,y>0

Determine which of the following equations is correct. - x∫14xdx=x \int \frac { 1 } { 4 x } d x =x∫4x1​dx=

Solve the differential equation. - xdydx=cos⁡xx2−3y,x>0x \frac { d y } { d x } = \frac { \cos x } { x ^ { 2 } } - 3 y , x > 0xdxdy​=x2cosx​−3y,x>0