Find the particular solution that satisfies the differential equation and the initial conditions.
F'(x) = 4x^2 f (-1)=3
F'(t)=10t-12t^3 f (3)=2
f' = 4x^2
f = 4/3 x^3 + C
Since f(-1)=3,
(4/3)(-1)+C = 3
C = 13/3
So,
f(x) = 4/3 x^3 + 13/3
do the other one in like wise.
To find the particular solution that satisfies the differential equation and the initial conditions, we need to solve the given differential equation and use the initial conditions to determine the constant(s) in the solution.
1) F'(x) = 4x^2, f(-1) = 3:
To solve this differential equation, we need to integrate both sides with respect to x:
∫ F'(x) dx = ∫ 4x^2 dx
Integrating, we get:
F(x) = (4/3)x^3 + C
Now, substituting the initial condition f(-1) = 3 into the equation:
F(-1) = (4/3)(-1)^3 + C = 3
Simplifying, we have:
-4/3 + C = 3
C = 3 + 4/3
C = 13/3
Therefore, the particular solution that satisfies the differential equation and the initial condition is:
F(x) = (4/3)x^3 + 13/3
2) F'(t) = 10t - 12t^3, f(3) = 2:
Again, we need to integrate both sides of the equation:
∫ F'(t) dt = ∫ (10t - 12t^3) dt
Integrating, we get:
F(t) = 5t^2 - 3t^4 + C
Now substitute the initial condition f(3) = 2 into the equation:
F(3) = 5(3)^2 - 3(3)^4 + C = 2
Simplifying, we have:
45 - 81 + C = 2
-36 + C = 2
C = 38
Therefore, the particular solution that satisfies the differential equation and the initial condition is:
F(t) = 5t^2 - 3t^4 + 38
To find the particular solution that satisfies the given differential equation and initial conditions, we will integrate the differential equation and use the initial condition to find the value of the constant of integration.
For the first differential equation, F'(x) = 4x^2, we can integrate both sides to find F(x):
∫ F'(x) dx = ∫ 4x^2 dx
F(x) = (4/3) x^3 + C
Now, we use the initial condition f(-1) = 3 to find the value of the constant C:
F(-1) = (4/3) (-1)^3 + C
3 = (-4/3) + C
C = 3 + (4/3)
C = 13/3
Therefore, the particular solution to the first differential equation that satisfies the initial condition f(-1) = 3 is:
F(x) = (4/3) x^3 + 13/3
For the second differential equation, F'(t) = 10t - 12t^3, we integrate both sides to find F(t):
∫ F'(t) dt = ∫ (10t - 12t^3) dt
F(t) = 5t^2 - 3t^4 + C
Using the initial condition f(3) = 2, we can find the value of the constant C:
F(3) = 5(3)^2 - 3(3)^4 + C
2 = 45 - 243 + C
C = 2 - 45 + 243
C = 200
Thus, the particular solution to the second differential equation that satisfies the initial condition f(3) = 2 is:
F(t) = 5t^2 - 3t^4 + 200