please help me out in calculus

Calculus
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#1.Write and then solve for y = f(x) the differential equation for the statemen "The rate of change of y with respect to x is inversely proportional to y^4."  #2.Solve the differential equation  for y = f(x) with the condition y(1) = 1.

Apr 29th, 2015

Rate of change = dy/dx

Equation: dy/dx = A/y^4 where A is an unknown proportionality constant.

Solving the diff. equation: y^4*dy=A*dx integrating: y^5/5=Ax+B, B is another constant relating to initial condition that do not affect the rate of change.

put y=1 at x=1 we get 1/5 =A+B. or:

y^5/5=Ax + 1/5-A or y^5=5A(x+1)-5  or y= 5th root[(5A(x+1)-5]

We cannot determine A completely unless we have one more data point. For example, if we are given also that y(0)=0 we can solve for A, B : B= 0 , A = 1/5.

Apr 29th, 2015

#2.Solve the differential equation dy/dx=y^2/x^3 for y = f(x) with the condition y(1) = 1.

Apr 29th, 2015

similarly as above: dy/y^2=dx/x^3  integrating -1/y = -1/(2*x^2) + A1=> 1/y=1/(2x^2) + A

find A: y(1)=1 : 1=1/2 + A => A=1/2

function 1/y=1/2(1/x^2+1)

 y= (x^2+1)/(2x^2)

Apr 29th, 2015

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