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