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mote1985 [20]
3 years ago
6

Find the solution of the initial value problem dy/dx=(-2x+y)^2-7 ,y(0)=0

Mathematics
1 answer:
Leokris [45]3 years ago
5 0

Substitute v(x)=-2x+y(x), so that \dfrac{\mathrm dv}{\mathrm dx}=-2+\dfrac{\mathrm dy}{\mathrm dx}. Then the ODE is equivalent to

\dfrac{\mathrm dv}{\mathrm dx}+2=v^2-7

which is separable as

\dfrac{\mathrm dv}{v^2-9}=\mathrm dx

Split the left side into partial fractions,

\dfrac1{v^2-9}=\dfrac16\left(\dfrac1{v-3}-\dfrac1{v+3}\right)

so that integrating both sides is trivial and we get

\dfrac{\ln|v-3|-\ln|v+3|}6=x+C

\ln\left|\dfrac{v-3}{v+3}\right|=6x+C

\dfrac{v-3}{v+3}=Ce^{6x}

\dfrac{v+3-6}{v+3}=1-\dfrac6{v+3}=Ce^{6x}

\dfrac6{v+3}=1-Ce^{6x}

v=\dfrac6{1-Ce^{6x}}-3

-2x+y=\dfrac6{1-Ce^{6x}}-3

y=2x+\dfrac6{1-Ce^{6x}}-3

Given the initial condition y(0)=0, we find

0=\dfrac6{1-C}-3\implies C=-1

so that the ODE has the particular solution,

\boxed{y=2x+\dfrac6{1+e^{6x}}-3}

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2 years ago
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vlada-n [284]

The slope-intercept form: y = mx + b

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b - y-intercept

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3 years ago
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2 years ago
An equilateral triangle has has an era of 43.5 square feet the height is 8.5 feet what is the Perimeter of the triangle
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Answer:

30.7 ft

Step-by-step explanation:

<em>An equilateral triangle with area 43.5 has a height of 8.68 not 8.5</em>

<em>I will do the problem as written even though it is not a possible equilateral triangle</em>

<em>---------------------------------------------</em>

a = (1/2)bh

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b = 43.5/4.25

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2 years ago
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