Question 1

Find all the curves  such that the tangent line at any point  on the curve has a y-intercept equal to  .

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Question 2

Solve the differential equation.

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Question 3

Solve the differential equation.

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Question 4

Consider the  circuit shown in the following figure.  At  the switch is closed and the current passes through the circuit.
The constant voltage  is the sum of the voltage  across the resistor and the voltage  across the inductor. 
A) Set up an initial value problem satisfied by  . 
B) Solve for the current  . 
C) What is the current after a long time?

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Question 5

Solve the initial value problems.
A)  
B)

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A)  
B)...

Question 6

Solve the initial value problem.

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Question 7

A certain chemical dissolves in water at a rate proportional to the product of the amount of chemical that had not yet been dissolved and the difference between its concentration in a saturated solution and its current concentration.
It is known that 100 g of the chemical are dissolved in 200 g of saturated solution. If 60 g of the chemical are added to 200 g of water, 20 g are dissolved in 2 h.
How many grams of the chemical are dissolved in 4 h?

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Question 8

An object of constant mass is projected away from the earth with initial velocity  in a direction perpendicular to the earth's surface. Assuming there is no air resistance and considering the variation of the earth's gravitational field as a function of the altitude  above the earth's surface, the velocity of the object satisfies the differential equation  , where  is the radius of the earth.  
A) Solve for the velocity  as a function of  . 
B) What is the maximum altitude  of the object if its initial velocity is  ? 
C) Find the escape velocity, that is, the least initial velocity for which the body will not return to the earth.
(Hint: find  such that  .)

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Question 9

Solve the differential equations using separation of variables.
A)  
B)

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Question 10

Find all the curves  with the following property:
the y-intercept of the tangent line at any point  on the curve is equal to  .

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Question 11

A conical tank filled with water has a height of 7 m and a top radius of 2 m. Water leaks through a hole of area  at the bottom. 
A) Find the water level  at time  . 
B) How long does it takes for the tank to empty?

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Question 12

Solve the differential equations.
A)  
B)  . (Hint: factor the denominator.)

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A)  
B) ...

Question 13

The horizontal cross sections at height  of a tank are discs of radius  . The height of the tank is 10 m.
The tank is filled with water, and the water drains through a square hole with a side of 10 cm at the bottom of the tank.
How long does it take for the tank to empty?

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Question 14

Solve the initial value problem.

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Question 15

A water tank is obtained by rotating the graph of  for  about the y-axis (assume length is measured in meters).
The tank is filled with water and water drains through a circular hole of radius
1 cm at the bottom of the tank.
How long does it take for the tank to empty?

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Question 16

Solve the initial value problems.
A)  
B)

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A)  
B)...

Question 17

Find all curves  such that the tangent line at any point  on the curve has a y-intercept that is equal to  .

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Question 18

Solve the initial value problem.

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Question 19

Solve the initial value problems.
A)  
B)

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A)  
B)...

Question 20

A tank in the shape of a prism and filled with water has a height of 5 m. The cross sections are equilateral triangles with a side of 1 m. Water leaks through a hole of area 0.01  at the bottom of the tank.  
A) Find the water level  at time  . 
B) How long does it take for the tank to empty?

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Quiz 10: Introduction to Differential Equations

Find all the curves such that the tangent line at any point on the curve has a y-intercept equal to .

Solve the differential equation.

Solve the differential equation.

Solve the initial value problems. A) B)

Solve the initial value problem.

Solve the differential equations using separation of variables. A) B)

Find all the curves with the following property: the y-intercept of the tangent line at any point on the curve is equal to .

A conical tank filled with water has a height of 7 m and a top radius of 2 m. Water leaks through a hole of area at the bottom. A) Find the water level at time . B) How long does it takes for the tank to empty?

Solve the differential equations. A) B) . (Hint: factor the denominator.)

The horizontal cross sections at height of a tank are discs of radius . The height of the tank is 10 m. The tank is filled with water, and the water drains through a square hole with a side of 10 cm at the bottom of the tank. How long does it take for the tank to empty?

Solve the initial value problem.

A water tank is obtained by rotating the graph of for about the y-axis (assume length is measured in meters). The tank is filled with water and water drains through a circular hole of radius 1 cm at the bottom of the tank. How long does it take for the tank to empty?

Solve the initial value problems. A) B)

Find all curves such that the tangent line at any point on the curve has a y-intercept that is equal to .

Solve the initial value problem.

Solve the initial value problems. A) B)

A tank in the shape of a prism and filled with water has a height of 5 m. The cross sections are equilateral triangles with a side of 1 m. Water leaks through a hole of area 0.01 at the bottom of the tank. A) Find the water level at time . B) How long does it take for the tank to empty?