The graph of a function f is shown. Use the differential equation and the given point to find an equation of the function. dy/dx=4x+(9x^2/(3x^3+1)^(3/2))
and the point (0,2) using the substitution method of integration.
y' = 4x+(9x^2/(3x^3+1)^(3/2))
y = 2x^2 - 2/√(3x^3+1) + C
Now plug in (0,2) to find C.
integrate using u = 3x^3+1, so
du = 9x^2 dx
To find an equation of the function given the differential equation and the point (0,2), we will use the substitution method of integration.
Step 1: Start with the given differential equation:
dy/dx = 4x + (9x^2 / (3x^3 + 1)^(3/2))
Step 2: Separate the variables by moving all terms with y to one side and all terms with x to the other side:
(3x^3 + 1)^(3/2) dy = (4x + 9x^2) dx
Step 3: Integrate both sides of the equation separately:
∫ (3x^3 + 1)^(3/2) dy = ∫ (4x + 9x^2) dx
Step 4: To integrate the left side, we need to substitute u = 3x^3 + 1. Taking the derivative of u with respect to x, we get du/dx = 9x^2. Rearranging, we have dx = du / (9x^2).
Substituting this into the integral for the left side, we get:
∫ (3x^3 + 1)^(3/2) dy = ∫ (4x + 9x^2) dx
∫ u^(3/2) dy = ∫ (4x + 9x^2) (du/9x^2)
Step 5: Simplify the integral on the right side:
∫ u^(3/2) dy = ∫ (4x + 9x^2) (du/9x^2)
∫ u^(3/2) dy = (4/9) ∫ dx + ∫ du
Step 6: Integrate the right side:
∫ u^(3/2) dy = (4/9) x + u + C
Step 7: Substitute back u = 3x^3 + 1 and the given point (0,2) for y and x, respectively:
u = 3(0)^3 + 1 = 1
y = 2
∫ u^(3/2) dy = (4/9) x + u + C
∫ 1^(3/2) dy = (4/9) (0) + 1 + C
∫ 1 dy = 1 + C
Step 8: Simplify and find the value of C:
y = y + C
2 = 2 + C
C = 0
Step 9: Final equation:
y = (4/9) x + u
Substituting back u = 3x^3 + 1, the equation becomes:
y = (4/9) x + 3x^3 + 1
To find an equation of the function given the differential equation and the point (0,2), we can solve the differential equation using the method of separation of variables.
Step 1: Rearrange the differential equation.
Start by rearranging the given differential equation dy/dx = 4x + (9x^2/(3x^3 + 1)^(3/2)).
dy = (4x + (9x^2/(3x^3 + 1)^(3/2))) dx.
Step 2: Separate the variables.
To begin separating the variables, move the dx term to the other side.
dy/dx = 4x + (9x^2/(3x^3 + 1)^(3/2)).
dy = (4x + (9x^2/(3x^3 + 1)^(3/2))) dx.
Step 3: Perform integration.
Now, integrate both sides of the equation with respect to their respective variables.
∫ dy = ∫ (4x + (9x^2/(3x^3 + 1)^(3/2))) dx.
Integrating the left side gives y + C1, where C1 is the constant of integration.
For the right side, we will integrate the two terms separately.
∫ 4x dx + ∫ (9x^2/(3x^3 + 1)^(3/2)) dx.
Step 4: Evaluate the integrals.
Integrating 4x dx gives 2x^2 + C2, where C2 is another constant of integration.
To integrate the second term, we can use a substitution method.
Let u = 3x^3 + 1.
Taking the derivative of u with respect to x, we get du/dx = 9x^2.
Rearranging this equation gives du = 9x^2 dx.
Now we can substitute these values back into the integral.
∫ 4x dx + ∫ (9x^2/(3x^3 + 1)^(3/2)) dx becomes:
∫ 4x dx + ∫ (1/u^(3/2)) du.
Integrating 4x dx will give 2x^2 + C2.
Integrating (1/u^(3/2)) du can be done by raising u^(-3/2) to -1/2 and multiplying by the reciprocal of -1/2.
This gives us -2/u^(1/2) + C3, where C3 is yet another constant of integration.
Therefore, the right side of the equation becomes 2x^2 - 2/(3x^3 + 1)^(1/2) + C3.
So our equation becomes y + C1 = 2x^2 - 2/(3x^3 + 1)^(1/2) + C3.
Step 5: Use the given point to find the specific equation.
Using the given point (0,2), we substitute x = 0 and y = 2 into the equation found in Step 4.
2 + C1 = 2(0)^2 - 2/(3(0)^3 + 1)^(1/2) + C3.
Simplifying further, we have:
2 + C1 = 0 - 2/(1)^(1/2) + C3.
2 + C1 = 0 - 2 + C3.
Rearranging the equation, we get:
C1 + 2 = -2 + C3.
Since C1 and C3 are arbitrary constants, we can equate them.
C1 + 2 = C3 - 2.
Simplifying further, we have:
C1 = C3 - 4.
Therefore, the equation of the function is:
y = 2x^2 - 2/(3x^3 + 1)^(1/2) + C - 4, where C represents any constant.