Solved on Feb 17, 2024

Solve the differential equation dxdt=5t1+5\frac{d x}{d t}=5 t^{-1}+5 for the given initial conditions.

STEP 1

Assumptions
1. We are given the differential equation dxdt=5t1+5\frac{d x}{d t}=5 t^{-1}+5.
2. We need to integrate the right-hand side with respect to tt to find x(t)x(t).

STEP 2

To solve the differential equation, we will integrate both sides with respect to tt.
dxdtdt=(5t1+5)dt\int \frac{d x}{d t} \, dt = \int (5 t^{-1}+5) \, dt

STEP 3

The integral of the left-hand side with respect to tt is simply x(t)x(t).
x(t)=(5t1+5)dtx(t) = \int (5 t^{-1}+5) \, dt

STEP 4

We can split the integral on the right-hand side into two separate integrals.
x(t)=5t1dt+5dtx(t) = \int 5 t^{-1} \, dt + \int 5 \, dt

STEP 5

Integrate the first term, which is a power of tt.
5t1dt=5t1dt=5lnt+C1\int 5 t^{-1} \, dt = 5 \int t^{-1} \, dt = 5 \ln |t| + C_1
Here, C1C_1 is the constant of integration.

STEP 6

Integrate the second term, which is a constant with respect to tt.
5dt=5t+C2\int 5 \, dt = 5t + C_2
Here, C2C_2 is another constant of integration.

STEP 7

Combine the results of the two integrals.
x(t)=5lnt+5t+Cx(t) = 5 \ln |t| + 5t + C
Here, C=C1+C2C = C_1 + C_2 is the combined constant of integration.

STEP 8

We have found the general solution to the differential equation.
x(t)=5lnt+5t+Cx(t) = 5 \ln |t| + 5t + C

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