(a) Given is the function h(x, y) = (y - 1)(x + 1). (1) Draw the level curves for h(x, y) = c forc = -1, 0, 1 in the xy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3), P₂ = (-1, 1) and P₁ = (-1,-1). Draw a direction in P3 which h neither increases nor decreases at P₁. Draw the gradient vector of h at P₂. Draw the direction of the steepest decrease at P3. Show your working and justify your choices. ) Consider the following second-order differential equation d2y dx² dy + +y = 0. dx (1) Find the general solution of the equation. (ii) Use the general solution from part (i) to evaluate lim y(x). #448 (iii) Find the particular solution satisfying y(0) = 0 and y (0)=√3. dy (0) = 0 and Calculate the particular solution of dy (0) -(0) = 0. d³y da³ =sin(x) with y(0) = 0, =
(a) Given is the function h(x, y) = (y - 1)(x + 1). (1) Draw the level curves for h(x, y) = c forc = -1, 0, 1 in the xy-plane. Label curves clearly with the appropriate value of c. Show all your working. (ii) In the drawing of the previous part, clearly mark points P₁ = (-1,3), P₂ = (-1, 1) and P₁ = (-1,-1). Draw a direction in P3 which h neither increases nor decreases at P₁. Draw the gradient vector of h at P₂. Draw the direction of the steepest decrease at P3. Show your working and justify your choices. ) Consider the following second-order differential equation d2y dx² dy + +y = 0. dx (1) Find the general solution of the equation. (ii) Use the general solution from part (i) to evaluate lim y(x). #448 (iii) Find the particular solution satisfying y(0) = 0 and y (0)=√3. dy (0) = 0 and Calculate the particular solution of dy (0) -(0) = 0. d³y da³ =sin(x) with y(0) = 0, =
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 1YT
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![(a) Given is the function h(x, y) = (y − 1)(x + 1).
(i) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the
xy-plane. Label curves clearly with the appropriate value of c.
Show all your working.
(c)
(ii) In the drawing of the previous part, clearly mark points
P₁ = (-1,3),
P₂ = (-₁1) and P3 = (-1,-1). Draw a direction in
which h neither increases nor decreases at P₁. Draw the
gradient vector of h at P₂. Draw the direction of the steepest
decrease at P3. Show your working and justify your choices.
(b) Consider the following second-order differential equation
dy+dy
(1) Find the general solution of the equation.
(ii) Use the general solution from part (i) to evaluate lim y(x).
#448
Calculate the particular solution of
d'y
(iii) Find the particular solution satisfying y(0) =
y (0) = √3.
d(0)
(0) = 0 and
+y=0.
(0)=
= 0.
d³y
= 0 and
=
=sin(x) with y(0) = 0,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab0e1c6e-2cd5-492d-90bb-1a9c33e07d97%2Fa6e0008c-75a4-4c74-b2ae-be4c8fb5809a%2Fp86tm7m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Given is the function h(x, y) = (y − 1)(x + 1).
(i) Draw the level curves for h(x, y) = c for c= -1,0, 1 in the
xy-plane. Label curves clearly with the appropriate value of c.
Show all your working.
(c)
(ii) In the drawing of the previous part, clearly mark points
P₁ = (-1,3),
P₂ = (-₁1) and P3 = (-1,-1). Draw a direction in
which h neither increases nor decreases at P₁. Draw the
gradient vector of h at P₂. Draw the direction of the steepest
decrease at P3. Show your working and justify your choices.
(b) Consider the following second-order differential equation
dy+dy
(1) Find the general solution of the equation.
(ii) Use the general solution from part (i) to evaluate lim y(x).
#448
Calculate the particular solution of
d'y
(iii) Find the particular solution satisfying y(0) =
y (0) = √3.
d(0)
(0) = 0 and
+y=0.
(0)=
= 0.
d³y
= 0 and
=
=sin(x) with y(0) = 0,
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