# math代做 – math 250 Online exam

### math 250 Online exam

math代做 – 这个题目属于一个express的代写任务, 涵盖了express/math等方面

``````Midterm Exam Winter 2023
A math 250 Online exam
``````

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1. Find the general solutions (implicit or explicit) to the following equations.

[4] (a) y(x)y(x) =x y^5

``````(b)
``````
``````dy
dx
``````

#### =

``````2 x^3 x y^2
x^2 y 2 y^3
``````

#### [4]

[4] (c) y 2 y+y=exsin(x)

1. Consider the DEy(x) = (y1)^2 (y2),

[4] (a) Sketch the direction field and determine the intervals of increase/decrease and intervals of concavity. Then sketch the solution curves for the initial conditions (ICs)y(0) = 0. 5 , 0. 5 , 1. 5 , 1. 9 , 2. 5.

[3] (b) Based on the direction field (a), determine how the behaviour of the so- lutionsyfor largex(i.e., asx+) depends on different initial values ofyatx= 0.

1. (a) Consider the following IVP,[3]
``````y(x) =
``````

#### 3

``````x y^2 /^5 , y(0) = 1
``````
``````Using the existence and uniqueness theorem, what can you predict about
the solution.
(b) Solve the IVP from part (a).[3]
``````
1. Consider the following IVP:[4]
``````d^2 x
dt^2
``````

#### =

``````g R^2
(x+R)^2
``````
``````, x(0) = 0 ,
``````
``````dx
dt
``````
``````(0) =v 0
``````
``````Heregis the gravitational acceleration (m/s^2 ),Ris the radius of the earth (m),
andv 0 is the initial velocity (m/s). Choose appropriate variables and write this
IVP in a non-dimensional form.
``````
1. Solve the following equations.

[4] (a) x yy(ln(x y)1) = 0

[3] (b) y= 1 + (y)^2