Part I. Finding solutions of a given differential equation. 1. Find - PDF document

Part I. Finding solutions of a given differential equation. 1. Find the real numbers r such that y = e rx is a solution of y ′′ − y ′ − 30 y = 0 2. Find the real numbers r such that y = e rx is a solution of y ′′ + 8 y ′ + 16 y = 0 1

3. Find the real numbers r such that y = e rx is a solution of y ′′ − 2 y ′ + 10 y = 0 4. Find the real numbers r such that y = x r is a solution of x 2 y ′′ − 5 xy ′ + 8 y = 0 5. Find the real numbers r such that y = x r is a solution of y ′′ − 5 x y ′ + 9 x 2 y = 0

Part II. Find the differential equa- tion for an n -parameter family of curves. Examples: 1. y 2 = Cx 3 − 2. 2. y = C 1 x 3 + C 2 x . 3. y = C 1 e − 2 x + C 2 xe − 2 x .

4. y 3 = C ( x − 2) 2 + 4 5. y = C 1 e 3 x cos 4 x + C 2 e 3 x sin 4 x . 6. y = C 1 e − 2 x + C 2 e 5 x . 7. y = C 1 + C 2 e 4 x + 2 x 8. y 3 = Cx 2 − 3 x

Part III. Identify each of the follow- ing first order differential equations. 1. x (1 + y 2 ) + y (1 + x 2 ) y ′ = 0. 2. xdy − 2 y x dx = x 3 e − x dx 3. ( xy + y ) y ′ = x − xy . 4. xy 2 dy dx = x 3 e y/x − x 2 y 5. y ′ = − 3 y x + x 4 y 1 / 3 .

6. (3 x 2 + 1) y ′ − 2 xy = 6 x . 7. x 2 y ′ = x 2 + 3 xy + y 2 8. x (1 − y ) + y (1 + x 2 ) dy dx = 0. 9. xy ′ = x 2 y + y 2 ln x .

Part IV. First order linear equa- tions; find general solution, solve an initial-value problem. 1. Find the general solution of x 2 dy − 2 xy dx = x 4 cos 2 x dx 2. Find the general solution of (1 + x 2 ) y ′ + 1 + 2 x y = 0

3. Find the general solution of xy ′ − y = 2 x ln x 4. Find the solution of the initial-value problem xy ′ + 3 y = e x y (1) = 2 x , 5. If y = y ( x ) is the solution of the initial-value problem y ′ + 3 y = 2 − 3 e − x , y (0) = 2 ,

then x →∞ y ( x ) = lim

Part V. Separable equations; find general solution, solve an initial-value problem. Examples: 1. Find the general solution of y ′ = xe x + y 2. Find the general solution of yy ′ = xy 2 − x − y 2 + 1

3. Find the general solution of ln x dy dx = y x 4. Find the solution of the initial-value problem y ′ = x 2 y − y y + 1 , y (3) = 1

Part VI. Bernoulli equations; find general solution. Examples: 1. Find the general solution of y ′ + xy = xy 3 2. Find the general solution of y ′ = 4 y + 2 e x √ y

3. Find the general solution of xy ′ + y = y 2 ln x

Part VII. Homogeneous equations; find general solution. Examples: 1. Find the general solution of y 2 y ′ = xy + y 2 2. Find the general solution of xy y ′ = x 2 e y/x + y 2

3. Find the general solution of � x 2 − y 2 y ′ = y + x

Part VIII. Applications Examples: 1. Given the family of curves y = Ce 2 x + 1 Find the family of orthogonal tra- jectories. 2. Given the family of curves y 2 = C ( x + 2) 3 − 2

Find the family of orthogonal tra- jectories. 3. Given the family of curves y 3 = Cx 2 + 2 Find the family of orthogonal tra- jectories. 4. A 200 gallon tank, initially full of water, develops a leak at the bot- tom. Given that 20% of the water

leaks out in the first 4 minutes, find the amount of water left in the tank t minutes after the leak develops if: (i) The water drains off a rate pro- portional to the amount of water present. (ii) The water drains off a rate pro- portional to the product of the time elapsed and the amount of water present.

(iii) The water drains off a rate pro- portional to the square root of the amount of water present. 5. A certain radioactive material is de- caying at a rate proportional to the amount present. If a sample of 100 grams of the material was present initially and after 2 hours the sam- ple lost 20% of its mass, find:

(a) An expression for the mass of the material remaining at any time t . (b) The mass of the material after 4 hours. (c) The half-life of the material. 6. A biologist observes that a certain bacterial colony triples every 4 hours and after 12 hours occupies 1 square

centimeter. Assume that the colony obeys the population growth law. (a) How much area did the colony occupy when first observed? (b) What is the doubling time for the colony? 7. A thermometer is taken from a room where the temperature is 72 o F to the outside where the temperature

32 o F . is After 1 / 2 minute, the 50 o F . Assume thermometer reads Newton’s Law of Cooling. (a) What will the thermometer read after it has been outside for 1 minute? (b) How many minutes does the ther- mometer have to be outside for it 35 o F ? to read

8. An advertising company designs a campaign to introduce a new prod- uct to a metropolitan area of pop- ulation M . Let P = P ( t ) de- note the number of people who be- come aware of the product by time t . Suppose that P increases at a rate proportional to the number of people still unaware of the prod- uct. The company determines that no one was aware of the product at the beginning of the campaign and

that 30% of the people were aware of the product after 10 days of ad- vertising. (a) Give the mathematical model (dif- ferential equation and initiation condition). (b) Determine the solution of the initial- value problem in (a). (c) Determine the value of the pro- portionality constant.

(d) How long does it take for 90% of the population to become aware of the product? 9. A disease is infecting a troop of 100 monkeys living in a remote preserve. Let M ( t ) be the number of sick monkeys t days after the outbreak. Suppose that the disease is spread- ing at a rate proportional to the product of the time elapsed and the

number of monkeys which have the disease. Suppose, also, that 20 monkeys had the disease initially, and that 30 monkeys were sick after 2 weeks. (a) Give the mathematical model (dif- ferential equation and initiation condition). (b) Determine the solution of the initial- value problem in (a).

(c) Determine the value of the pro- portionality constant. (d) How long does it take for 90% of the population to become aware of the product?

Part IX. Second order linear equa- tions; general theory. Examples: y 1 ( x ) = e 2 x , y 2 ( x ) = e − 2 x � � 1. is a fun- damental set of solutions of a linear homogeneous differential equation. What is the equation? y 1 ( x ) = x, y 2 ( x ) = x 3 � � 2. is a funda- mental set of solutions of a linear

homogeneous differential equation. What is the equation? 3. Given the differential equation x 2 y ′′ − 2 x y ′ − 10 y = 0 (a) Find two values of such that r y = x r is a solution of the equa- tion. (b) Determine a fundamental set of solutions and give the general so-

lution of the equation. (c) Find the solution of the equa- tion satisfying the initial condi- y (1) = 6 , y ′ (1) = 2. tions 4. Given the differential equation � 6 � 12 � � y ′ + y ′′ − y = 0 x 2 x (a) Find two values of such that r y = x r is a solution of the equa- tion.

(b) Determine a fundamental set of solutions and give the general so- lution of the equation. (c) Find the solution of the equa- tion satisfying the initial condi- y (1) = 2 , y ′ (1) = − 1. tions (d) Find the solution of the equa- tion satisfying the initial condi- y (2) = y ′ (2) = 0. tions

Part X. Homogeneous equations with constant coefficients. Examples: 1. Find the general solution of y ′′ + 10 y ′ + 25 y = 0 2. Find the general solution of y ′′ − 8 y ′ + 15 y = 0

3. Find the general solution of y ′′ + 4 y ′ + 20 y = 0 4. Find the solution of the initial-value problem: y ′′ − 2 y ′ +2 y = 0; y (0) = − 1 , y ′ (0) = − 1 5. Find the solution of the initial-value problem: y ′′ +4 y ′ +4 y = 0; y ( − 1) = 2 , y ′ ( − 1) = 1

6. Find the general solution of y ′′ − 2 α y ′ + α 2 y = 0 , a constant. α 7. Find the general solution of y ′′ − 2 α y ′ + ( α 2 + β 2 ) y = 0 , α, β constants. y = 2 e 3 x − 5 e − 4 x 8. The function is a solution of a second order linear

differential equation with constant coefficients. What is the equation? y = 7 e − 3 x cos 2 x 9. The function is a solution of a second order linear differential equation with constant coefficients. What is the equation? 10. Find a second order linear homo- geneous differential equation with constant coefficients that has y = e − 4 x as a solution.

y = 6 xe 4 x 11. The function is a solu- tion of a second order linear differ- ential equation with constant coef- ficients. What is the equation?