INTRODUCTION TO DIFFERENTIAL EQUATIONS 20 CHAPTER 1 In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP. 3y2/3, y(0) = 0 15. y' 16. ху' — 2у, у(0) — 0 In Problems 17–24 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (xo, yo) in the region. dy dy = y2/3 17. dx 18. Vху dx dy 19. х dx dy 20. у3 х dx y²)y' = x² 22. (1 + y³)y' = x² 21. (4 23. (х2 + у?)у' у? 24. (у — х)у' %— у+x In Problems 25–28 determine whether Theorem 1.2.1 guarantees that the differential equation y' = Vy2 – 9 possesses a unique solution through the given point. 25. (1, 4) 26. (5, 3) 28. (–1, 1) 27. (2, –3) 29. (a) By inspection find a one-parameter family of solutions of the differential equation xy' = y. Verify that each member of the family is a solution of the initial-value problem xy' = y, y(0) = 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy' = y would have a unique solution through a point (xo, yo) in R. (c) Verify that the piecewise-defined function |0, х<0 y = satisfies the condition y(0) function is also a solution of the initial-value problem in = 0. Determine whether this part (a). 30. (a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y². (b) Since f(x, y) erywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem v' = 1 + v², v(0) = 0. Even though xn = 0 is in = 1 + y? and af/əy = 2y are continuous ev-
INTRODUCTION TO DIFFERENTIAL EQUATIONS 20 CHAPTER 1 In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP. 3y2/3, y(0) = 0 15. y' 16. ху' — 2у, у(0) — 0 In Problems 17–24 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (xo, yo) in the region. dy dy = y2/3 17. dx 18. Vху dx dy 19. х dx dy 20. у3 х dx y²)y' = x² 22. (1 + y³)y' = x² 21. (4 23. (х2 + у?)у' у? 24. (у — х)у' %— у+x In Problems 25–28 determine whether Theorem 1.2.1 guarantees that the differential equation y' = Vy2 – 9 possesses a unique solution through the given point. 25. (1, 4) 26. (5, 3) 28. (–1, 1) 27. (2, –3) 29. (a) By inspection find a one-parameter family of solutions of the differential equation xy' = y. Verify that each member of the family is a solution of the initial-value problem xy' = y, y(0) = 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy' = y would have a unique solution through a point (xo, yo) in R. (c) Verify that the piecewise-defined function |0, х<0 y = satisfies the condition y(0) function is also a solution of the initial-value problem in = 0. Determine whether this part (a). 30. (a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y². (b) Since f(x, y) erywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem v' = 1 + v², v(0) = 0. Even though xn = 0 is in = 1 + y? and af/əy = 2y are continuous ev-
Calculus: Early Transcendentals
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:INTRODUCTION TO DIFFERENTIAL EQUATIONS
20
CHAPTER 1
In Problems 15 and 16 determine by inspection at least two solutions
of the given first-order IVP.
3y2/3, y(0) = 0
15. y'
16. ху' — 2у, у(0) — 0
In Problems 17–24 determine a region of the xy-plane for which
the given differential equation would have a unique solution whose
graph passes through a point (xo, yo) in the region.
dy
dy
= y2/3
17.
dx
18.
Vху
dx
dy
19. х
dx
dy
20.
у3 х
dx
y²)y' = x²
22. (1 + y³)y' = x²
21. (4
23. (х2 + у?)у'
у?
24. (у — х)у' %— у+x
In Problems 25–28 determine whether Theorem 1.2.1 guarantees that
the differential equation y' = Vy2 – 9 possesses a unique solution
through the given point.
25. (1, 4)
26. (5, 3)
28. (–1, 1)
27. (2, –3)
29. (a) By inspection find a one-parameter family of solutions of
the differential equation xy' = y. Verify that each member
of the family is a solution of the initial-value problem
xy' = y, y(0) = 0.
(b) Explain part (a) by determining a region R in the xy-plane
for which the differential equation xy' = y would have a
unique solution through a point (xo, yo) in R.
(c) Verify that the piecewise-defined function
|0, х<0
y =
satisfies the condition y(0)
function is also a solution of the initial-value problem in
= 0. Determine whether this
part (a).
30. (a) Verify that y = tan (x + c) is a one-parameter family of
solutions of the differential equation y' = 1 + y².
(b) Since f(x, y)
erywhere, the region R in Theorem 1.2.1 can be taken to
be the entire xy-plane. Use the family of solutions in part
(a) to find an explicit solution of the first-order initial-value
problem v' = 1 + v², v(0) = 0. Even though xn = 0 is in
= 1 + y? and af/əy = 2y are continuous ev-
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