ENGR 213 ASSIGNMENT ASSIGNMENT 1 STUDY NOTES CONCORDIA UNIVERSITY
1. (1 point) It can be helpful to classify a differential equa- (a) The order of this differential equation is .
tion, so that we can predict the techniques that might help us to
find a function which solves the equation. Two classifications (b) The equation is [Choose/Linear/Nonlinear].
are the order of the equation – (what is the highest number of
derivatives involved) and whether or not the equation is linear 5. (1 point) Find all values of k for which the function
y = sin(kt) satisfies the differential equation y′ + 7y = 0.
. Linearity is important because the structure of the the family
Sep- arate your answers by commas. Hint: There are more than
of solutions to a linear equation is fairly simple. Linear
2 values of k
equations can usually be solved completely and explicitly.
Determine whether or not each equation is linear:
6. (1 point) Match the following differential equations with
their solutions.
? 1. y′ − y + t 2 = 0
The symbols A, B, C in the solutions stand for arbitrary con-
? 2. y′ −y + y2 = 0 stants.
d2y + dy + 2y sin t You must get all of the answers correct to receive credit.
? =
3. t dt
2 2
dt
d2y sin t y sin t d2y
? 4. 2 (= + )
dt 1. + 25y = 0
+ dx2
2. (1 point) In problems below, (a) identify the independent dy −2xy
2. dx = x2 5y2
variable and the dependent variable of each equation (use ’t’
d2y dy−
for the independent variable if an independent variable is not
given explicitly); (b) give the order of each differential 3.dy 2 + 6 + 9y = 0
4. dx = 10xydx
equation (enter ’1’ for first order, ’2’ for second order and so dx
dy
on; do not include the quotes); and (c) state whether the 5. + 9x2y = 9x2
equation is linear or nonlinear. If your answer to (c) is dx
nonlinear, make sure that you can explain why this is true. A. 3yx2 5y3 = C
−
B. y = A cos(5x) + B sin(5x)
C. y = Ae−3x + Bxe−3x
D. y = Ae5x2
equation (c) linEe.ary/
(a) independent (a) dependent (b) order
y′ = y n=onClein−e3axr3
− +1
x2 xy′ =
2y [?/linear/nonlinear]
7 . ( 1 poin t ) W h ich of the following functions
x′ + 5x = e−x are
[?/l solutions
in e a r/n li n e ar ]
o f t h e d i f fe r e n ti a l e quation y′ − 10y′ + 25y =
[0?? /l i n ea r / n o n l in e a r ]
3. (1 point) Determine the order of the given differential • A. y(x) = e−5x
equation and state whether the equation is linear or nonlinear. • B. y(x) = 5xe−5x
• C. y(x) = xe5x
(sin θ)y′ ′ − (cos θ)y′ = 9 • D. y(x) = e5x
• E. y(x) = 0
(a) The order of this differential equation is . • F. y(x) = x2e5x
• G. y(x) = 5x
(b) The equation is [Choose/Linear/Nonlinear].
8. (1 point) If y = e3t is a solution to the differential equation
d2y dy
4. (1 point) Determine the order of the given differential d2u du
equation and state whether the equation is linear or nonlinear.
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1. (1 point) It can be helpful to classify a differential equa- (a) The order of this differential equation is .
tion, so that we can predict the techniques that might help us to
find a function which solves the equation. Two classifications (b) The equation is [Choose/Linear/Nonlinear].
are the order of the equation – (what is the highest number of
derivatives involved) and whether or not the equation is linear 5. (1 point) Find all values of k for which the function
y = sin(kt) satisfies the differential equation y′ + 7y = 0.
. Linearity is important because the structure of the the family
Sep- arate your answers by commas. Hint: There are more than
of solutions to a linear equation is fairly simple. Linear
2 values of k
equations can usually be solved completely and explicitly.
Determine whether or not each equation is linear:
6. (1 point) Match the following differential equations with
their solutions.
? 1. y′ − y + t 2 = 0
The symbols A, B, C in the solutions stand for arbitrary con-
? 2. y′ −y + y2 = 0 stants.
d2y + dy + 2y sin t You must get all of the answers correct to receive credit.
? =
3. t dt
2 2
dt
d2y sin t y sin t d2y
? 4. 2 (= + )
dt 1. + 25y = 0
+ dx2
2. (1 point) In problems below, (a) identify the independent dy −2xy
2. dx = x2 5y2
variable and the dependent variable of each equation (use ’t’
d2y dy−
for the independent variable if an independent variable is not
given explicitly); (b) give the order of each differential 3.dy 2 + 6 + 9y = 0
4. dx = 10xydx
equation (enter ’1’ for first order, ’2’ for second order and so dx
dy
on; do not include the quotes); and (c) state whether the 5. + 9x2y = 9x2
equation is linear or nonlinear. If your answer to (c) is dx
nonlinear, make sure that you can explain why this is true. A. 3yx2 5y3 = C
−
B. y = A cos(5x) + B sin(5x)
C. y = Ae−3x + Bxe−3x
D. y = Ae5x2
equation (c) linEe.ary/
(a) independent (a) dependent (b) order
y′ = y n=onClein−e3axr3
− +1
x2 xy′ =
2y [?/linear/nonlinear]
7 . ( 1 poin t ) W h ich of the following functions
x′ + 5x = e−x are
[?/l solutions
in e a r/n li n e ar ]
o f t h e d i f fe r e n ti a l e quation y′ − 10y′ + 25y =
[0?? /l i n ea r / n o n l in e a r ]
3. (1 point) Determine the order of the given differential • A. y(x) = e−5x
equation and state whether the equation is linear or nonlinear. • B. y(x) = 5xe−5x
• C. y(x) = xe5x
(sin θ)y′ ′ − (cos θ)y′ = 9 • D. y(x) = e5x
• E. y(x) = 0
(a) The order of this differential equation is . • F. y(x) = x2e5x
• G. y(x) = 5x
(b) The equation is [Choose/Linear/Nonlinear].
8. (1 point) If y = e3t is a solution to the differential equation
d2y dy
4. (1 point) Determine the order of the given differential d2u du
equation and state whether the equation is linear or nonlinear.
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