Solve the following differential equations. Use the method of Bernoulli’s

Carole Yarbrough 2021-12-20 Answered
Solve the following differential equations. Use the method of Bernoulli’s Equation. x(2x3+y)dy6y2dx=0
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Bob Huerta
Answered 2021-12-21 Author has 41 answers
Step 1
General Bernoullis

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Mason Hall
Answered 2021-12-22 Author has 36 answers
Solve xdy(x)dx(2x3+y(x))+6y(x)2=0:
Solve for dy(x)dx:
dy(x)dx=6y(x)2x(2x3+y(x))
Write the differential equation in terms of x. Since dydxdxdy=1,dy(x)dx=1dx(y)dy:
1dx(y)dy=6y2(y+2x(y)3)x(y)
Raise both sides to the power -1 and expand:
dx(y)dy=x(y)43y2+xy6y
Subtract xy6y from both sides:
dx(y)dyxy6y=x(y)43y2
Divide both sides by 13x(y)4:
3dx(y)dyx(y)4+12yx(y)3=1y2
Let v(y)=1x(y)3, which gives dv(y)dy=3dx(y)dyx(y)4

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RizerMix
Answered 2021-12-29 Author has 438 answers

The equation 6y2dxx(2x3+y)dy=0 , for y0 can be written as the following Bernulli equation in the unknown x(y)
dx/dyx/6y=x4/3y2. To reduce the equation to a linear one take x=V(y)1/3. Obtain V+V/2y=1/y2. The integrating factor is y1/2 and the solution is V=(y1/2)(Integral of (1/y2)(y1/2)dy+C)=2/y+C/y1/2.
Obtain the solution as x3=y/(2+Cy1/2) or (2x3y)2=Kyx6,K=C2

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