How do you use the differential equation #dy/dx=4x+(9x^2)/(3x^3+1)^(3/2)# to find the equation of the function given point (0,2)?

Question

William Abdalla · Accepted Answer

#y = 2x^2 - 2/sqrt(3x^3 + 1) + 4#

This is a separable differential equation.

#dy/dx = 4x + (9x^2)/(3x^3 + 1)^(3/2)#

#dy = (4x + (9x^2)/(3x^3 + 1)^(3/2))dx#

Integrate both sides.

#int(dy) = int 4x + (9x^2)/(3x^3 + 1)^(3/2)dx#

We can integrate the right-hand side using the rule #intx^ndx= x^(n + 1)/(n + 1) + C#, where #n != -1# and a substitution. Let #u = 3x^3 + 1#. Then #du = 9x^2dx# and #dx = (du)/(9x^2)#.

#y = 2x^2 + int (9x^2)/u^(3/2) * (du)/(9x^2)#

#y = 2x^2 + intu^(-3/2)du#

Integrate using the rule above:

#y = 2x^2 + -2/u^(1/2)#

#y = 2x^2 -2/sqrt(3x^3 + 1) + C#

The last step is to find the value of #C#. We know that when #x = 0#, #y = 2#. So:

#2 = 2(0)^2 - 2/sqrt(3(0)^3 + 1)+ C#

#2 = -2 + C#

#C = 4#

The solution to the differential equation is therefore #y = 2x^2 - 2/sqrt(3x^3 + 1) + 4#. Differentiating will yield the original equation.

Hopefully this helps!

Samantha Adkison · Answer

undefined To find the equation of the function given the point (0,2) using the differential equation dy/dx=4x+(9x^2)/(3x^3+1)^(3/2), follow these steps:

1. Integrate both sides of the given differential equation with respect to x to find the general solution.
2. Use the given point (0,2) to determine the constant of integration.
3. Substitute the constant of integration and the given point into the general solution to find the specific equation of the function.