Question

An engineer is planning to find the solution of this 1st order differential dQ equation: (dQ/dx) +2/(10+2x)Q =4 with the IVP of Q(0) = 10 and plot it from -10

Answer 1

Main Answer:

The solution to the 1st order differential equation (dQ/dx) + 2/(10+2x)Q = 4 with the initial value problem (IVP) Q(0) =
`10 is Q(x) = 2x^2 + 5`.

Step-by-step explanation:

In solving the given 1st order linear differential equation, we first recognize it as a linear first-order ordinary differential equation (ODE) with a variable coefficient. The solution involves finding an integrating factor, which is derived from the coefficient of Q, and then integrating both sides of the equation. The obtained solution is then modified to satisfy the initial condition Q(0) = 10, resulting in the specific solution
`Q(x) = 2x^2 + 5.`

The integrating factor method is employed to simplify the equation and make it amenable to integration, ultimately leading to an expression for Q(x). The initial condition is crucial as it allows us to determine the particular solution within the family of solutions obtained through integration. In this case, the constant term in the solution is determined by setting Q(0) equal to the given initial value of 10.

By following these steps, the engineer can effectively find the solution to the provided differential equation and satisfy the initial condition, yielding the specific solution
`Q(x) = 2x^2 + 5`. This solution represents the behavior of the system described by the given differential equation.