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love history [14]
3 years ago
13

A function y(t) satisfies the differential equation dy dt = y 4 − 6y 3 + 5y 2 . (a) What are the constant solutions of the equat

ion? (Recall that these have the form y = C for some constant, C.) (b) For what values of y is y increasing? (c) For what values of y is y decreasing?
Mathematics
2 answers:
Elza [17]3 years ago
8 0

Answer:

a) y = 0 , 5,1

b) y ⊂ (- ∞,0) ∪ (0,1)∪(5,∞)

Step-by-step explanation:

Given data:

differential equation is given as

\frac{dy}[dt} = y^4 -6y^3+ 5y^2

a) constant solution

y^4 -6y^3+ 5y^2 = 0

taking y^2 from all part

y^2(y^2 - 6y -5) = 0

solution of above equation is

y = 0 , 5,1

b) for which value y is increasing

\frac{dy}{dt}  > 0

y^2(y - 5) (y -1) > 0

y ⊂ (- ∞,0) ∪ (0,1)∪(5,∞)

Alexxandr [17]3 years ago
5 0

Answer:

Hence increasing in  (-\infty,0) U (1,5)

c) Decreasing in (0,1)

Step-by-step explanation:

Given that y(t) satisfies the differential equation

\frac{dy}{dt} =y^4-6y^2+5y^2\\=y^2(y^2-6y+5)\\=y^2(y-1)(y-5)

Separate the variables to have

\frac{dy}{y^2(y-1)(y-5)} =dt

Left side we can resolve into partial fractions

Let \frac{1}{y^2(y-1)(y-5)} =\frac{A}{y} +\frac{B}{y^2}+\frac{C}{y-1} \frac{D}{y-5}

Taking LCD we get

1= Ay(y-1)(Y-5) +B(y-1)(y-5)+Cy^2 (y-5)+Dy^2 (y-1)\\Put y =1\\1 =  -4C\\Put y =5\\ 1 = 25(4)D\\Put y =0\\1=5B\\

By equating coeff of y^3 we have

A+C+D=0

C=\frac{-1}{4} \\D=\frac{1}{100} \\B =\frac{1}{5} \\A = -C-D = \frac{6}{25}

Hence left side =

\frac{6}{25y} +\frac{1}{5y^2}+\frac{-1}{4(y-1)}+ \frac{1}{100(y-5)}=dt\\\frac{6}{25}ln y -\frac{1}{5y}-\frac{1}{4}ln|(y-1)| +\frac{1}{100}ln|y-5| = t+C

b) y is increasing whenever dy/dt>0

dy/dt =0 at points y =0, 1 and 5

dy/dt >0 in (-\infty,0) U (1,5)

Hence increasing in  (-\infty,0) U (1,5)

c) Decreasing in (0,1)

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