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# 2 1. Solve the following differential equation: y′′(x) + 6y′(x) + 25y(x) = 0. 3 2. Consider a linear differential equation ay′′ + by′ + cy = f(x) with constant coefficients. Let y1(x), y2(x) be...

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1. Solve the following differential equation:
y′′(x) + 6y′(x) + 25y(x) = 0.
3
2. Consider a linear differential equation
ay′′ + by′ + cy = f(x)
with constant coefficients. Let y1(x), y2(x) be solutions to the homogeneous equation
ay′′ + by′ + cy = 0
and let yp(x) be a particular solution
Prove that C1y1 + C2y2 + yp is a solution for any choice of constants C1, C2.
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3. A swimming pool whose volume is 10,000 gal contains water that is 0.01% chlorine. Starting at t = 0, city wate
containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate.
What is the percentage of chlorine in the pool after 1 h? When will the pool water be 0.002% chlorine?
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4. There are infinitely many different solutions to y′′ − y = 0 with conditions
y(0) = 0 and y′(0) = 1.
This is TRUE / FALSE.
Reason:
5. Solve the following differential equation:
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y′′′ + y′′ + 4y′ + 4y = 0.
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6. Find the general solution of the differential equation:
y′′′ − y′′ + y′ − 1 = 0
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7.
(a) Draw a phase line diagram for the differential equation
dx/dt = (2 − 5x)3(1 − 2x)(1 − 4x2).
Expected in the diagram are equili
ium points and signs of x′ (or flow direction
markers < and >).
(b) Draw a phase diagram using the phase line diagram of (a). Add these labels as
appropriate: funnel, spout, node, stable, unstable. Show at least 10 threaded curves. A
direction field is not required.

## Solution

Rajeswari answered on Feb 19 2022
a) The percentage of chlorine after 1 hour is 0.00973%.
) The pool water will habe a concentration of 0.002% chlorine at 4394 minutes (or 73.24 hours).
Step-by-step explanation:
We can define as X(t) the amount of chlorine that is in the pool at time t.
Then, the rate of change of X can be...
SOLUTION.PDF