Find the particular solution that satisfies the differential equation and the initial conditions.
f''(x) = x^2 f'(0) = 8, f(o)=4
f'(t) 10t - 12t^3, f(3) =2
f''(x) = x^2 f'(0) = 8, f(o)=4
if f''(x) = x^2, then
f'(x) = (1/3)x^3 + c
for the given:
8 = (1/3)(0) + c ---> c = 8 , and
f'(x) = (1/3)x^3 + 8
f(x) = (1/12)x^4 + 8x + k
f(0) = 4
4 = (1/12)(0) + 8(0) + k ----> k = 4
f(x) = (1/12)x^4 + 8x + 4
do the 2nd question in the same way
To find the particular solution that satisfies the given differential equation and initial conditions, we can use the method of integrating factors.
1. Start by rewriting the given differential equation as a first-order system:
f''(x) = x^2
Let y = f'(x), so we have:
y' = x^2
2. Integrate both sides of the equation with respect to x to solve for y:
∫ y' dx = ∫ x^2 dx
y = (1/3) x^3 + C1
3. Now we have an expression for y. To find f(x), we integrate y with respect to x:
f(x) = ∫ [ (1/3) x^3 + C1 ] dx
f(x) = (1/12) x^4 + C1x + C2
4. Apply the initial conditions to find the values of the constants C1 and C2.
Given: f'(0) = 8 and f(0) = 4.
From f'(x), we can find the derivative of f(x):
f'(x) = (1/3) x^3 + C1
Substituting x = 0, we get:
f'(0) = (1/3) (0)^3 + C1
8 = C1
From f(x), we can find the value of f(0):
f(0) = (1/12) (0)^4 + C1(0) + C2
4 = C2
5. Substitute the values of C1 and C2 into the expression for f(x):
f(x) = (1/12) x^4 + 8x + 4
Therefore, the particular solution that satisfies the given differential equation and initial conditions is:
f(x) = (1/12) x^4 + 8x + 4.