calculus. suppose dR/dt= (d/R)^(2) and P(1) 4. separate the differential equation, integrate both sides
To solve the differential equation, we'll separate the variables and integrate both sides of the equation.
Given: `dR/dt = (d/R)^2`, where `P(1) = 4`.
First, let's rewrite the equation in a more convenient form:
`dR/(d/R)^2 = dt`
Now, we'll integrate both sides of the equation:
∫`dR/(d/R)^2` = ∫`dt`
To simplify the left side, let's rewrite it using the properties of exponents and the chain rule for differentiation:
∫`R^2 dR`
Now we can integrate both sides.
∫`R^2 dR` = ∫`dt`
To integrate the right side, we simply have `∫dt = t + C`, where 'C' is the constant of integration.
For the left side integral, we'll use the power rule of integration, which states that ∫`x^n dx = (1/(n+1)) * x^(n+1) + C`.
Using this rule, we have:
(1/3) * R^3 + C₁ = t + C₂
Now we have two constants of integration, C₁ and C₂, which we'll address later.
To solve for R, we can rearrange the equation:
(1/3) * R^3 = t + (C₂ - C₁)
To simplify it further, we can combine the constants:
C = C₂ - C₁, where C is the combined constant.
Then we have:
(1/3) * R^3 = t + C
Finally, to solve for R, we can take the cube root on both sides:
R = (3(t + C))^(1/3)
Now, we have obtained the general solution for R in terms of t and the combined constant C.
To find the particular solution, we need to use the initial condition:
P(1) = 4
From the given information, we know that P(1) = R when t = 1. Substituting these values into the equation, we get:
4 = (3(1 + C))^(1/3)
To solve for C, we need to isolate it by raising both sides to the power of 3:
64 = 3(1 + C)
Now, let's solve for C:
64/3 = 1 + C
C = 64/3 - 1
C = 61/3
Substituting this value of C back into the equation, we have:
R = (3(t + 61/3))^(1/3)
Therefore, the particular solution to the differential equation is:
R = (3(t + 61/3))^(1/3)
To separate the differential equation, we need to isolate the terms involving dR and R on separate sides of the equation.
The given differential equation is:
dR/dt = (d/R)^2
Let's start by multiplying both sides of the equation by R^2:
R^2 * dR/dt = d^2
Next, we can rewrite the left side using the chain rule:
d(R^3/3)/dt = d^2
Now, we can integrate both sides with respect to t:
∫d(R^3/3)/dt dt = ∫d^2 dt
Integrating the left side gives us:
R^3/3 = ∫d^2 dt
To integrate the right side, we can write d^2 as (dR)^2:
R^3/3 = ∫(dR)^2 dt
Now, we have successfully separated the differential equation and integrated both sides.