Find the particular solution that satisfies the differential equation and the initial condition.
f ''(x) = x^2, f '(0) = 8, f(0) = 8
f (x) = ?
d^2y/dx^2 = x^2
dy/dx = (1/3)x^3 + c
8 = 0 + c so c = 8
or
dy/dx = (1/3)x^3 + 8
y = (1/12)x^4 + 8 x + k
8 = 0 + k
so k = 8
y = (1/12)x^4 + 8 x + 8
To solve the differential equation f ''(x) = x^2, we need to integrate twice.
Integrating x^2 once gives us:
f '(x) = (1/3)x^3 + C1, where C1 is the constant of integration.
Next, we integrate f '(x) to find f(x):
f(x) = (1/12)x^4 + C1x + C2, where C2 is the constant of integration.
Now we will use the initial conditions to find the particular solution.
Given f(0) = 8, substituting x=0 in the expression for f(x), we get:
8 = (1/12)(0)^4 + C1(0) + C2
8 = C2
Next, given f '(0) = 8, substituting x=0 in the expression for f '(x), we get:
8 = (1/3)(0)^3 + C1
8 = C1
Therefore, the particular solution that satisfies the differential equation and the initial conditions is:
f(x) = (1/12)x^4 + 8x + 8
To find the particular solution that satisfies the given differential equation and initial conditions, we can follow these steps:
Step 1: Solve the differential equation.
To solve the differential equation f ''(x) = x^2, we need to integrate both sides twice.
Integrating the equation once gives us:
f '(x) = (x^3)/3 + C₁,
where C₁ is a constant of integration.
Integrating the equation again gives us:
f(x) = (x^4)/12 + C₁x + C₂,
where C₂ is another constant of integration.
Step 2: Apply the initial conditions.
We are given two initial conditions:
f '(0) = 8 and f(0) = 8.
Using the first initial condition, we can substitute x = 0 into the equation for f '(x):
f '(0) = (0^3)/3 + C₁ = C₁.
Therefore, C₁ = 8.
Using the second initial condition, we can substitute x = 0 into the equation for f(x):
f(0) = (0^4)/12 + C₁(0) + C₂ = C₂.
Therefore, C₂ = 8.
Step 3: Substitute the values of C₁ and C₂ into the equation for f(x).
Using the values of C₁ = 8 and C₂ = 8, we can rewrite the equation for f(x) as:
f(x) = (x^4)/12 + 8x + 8.
Therefore, the particular solution f(x) that satisfies the given differential equation and initial conditions is:
f(x) = (x^4)/12 + 8x + 8.