Find the particular solution that satisfies the differential equation and the initial condition.

fā€‰''(x) = x^2, fā€‰'(0) = 8, f(0) = 8
f (x) = ?

User Icon for Damon Damon answered
8 years ago

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

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

To solve the differential equation fā€‰''(x) = x^2, we need to integrate twice.

Integrating x^2 once gives us:
fā€‰'(x) = (1/3)x^3 + C1, where C1 is the constant of integration.

Next, we integrate fā€‰'(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 fā€‰'(0) = 8, substituting x=0 in the expression for fā€‰'(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

User Icon for Explain Bot Explain Bot answered
11 months ago

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 fā€‰''(x) = x^2, we need to integrate both sides twice.

Integrating the equation once gives us:
fā€‰'(x) = (x^3)/3 + Cā‚,

where Cā‚ is a constant of integration.

Integrating the equation again gives us:
f(x) = (x^4)/12 + Cā‚x + Cā‚‚,

where Cā‚‚ is another constant of integration.

Step 2: Apply the initial conditions.
We are given two initial conditions:
fā€‰'(0) = 8 and f(0) = 8.

Using the first initial condition, we can substitute x = 0 into the equation for fā€‰'(x):
fā€‰'(0) = (0^3)/3 + Cā‚ = Cā‚.

Therefore, Cā‚ = 8.

Using the second initial condition, we can substitute x = 0 into the equation for f(x):
f(0) = (0^4)/12 + Cā‚(0) + Cā‚‚ = Cā‚‚.

Therefore, Cā‚‚ = 8.

Step 3: Substitute the values of Cā‚ and Cā‚‚ into the equation for f(x).
Using the values of Cā‚ = 8 and Cā‚‚ = 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.