Question

Consider the differential equation dx/dt​=eᵗ/eᵗ+3 (a) State the form of the differential equation and hence state which of the methods described in Unit 8 for finding solutions of differential equations you would use to solve this equation. (b) Find the general solution of the differential equation in explicit form. (c) Hence find the particular solution of the differential equation that satisfies the initial condition x(0)=17.

Answer 1

To solve the differential equation dx/dt = eᵗ/(eᵗ+3), we use the method of integrating factors. The general solution is x = (1/2)e²ᵗ + C, and the particular solution that satisfies the initial condition x(0) = 17 is x = (1/2)e²ᵗ + 17 - (1/2).

Step-by-step explanation:

To solve the differential equation dx/dt = eᵗ/(eᵗ+3), we can rewrite it as dx/(eᵗ+3) = eᵗ dt. This is a first-order linear differential equation, which can be solved using the method of integrating factors.

Multiplying both sides of the equation by eᵗ+3, we have dx = e²ᵗ dt. Integrating both sides, we get x = (1/2)e²ᵗ + C, where C is the constant of integration.

To find the particular solution that satisfies the initial condition x(0) = 17, we substitute x = 17 and t = 0 into the equation and solve for C. We have 17 = (1/2)e⁰ + C, which simplifies to C = 17 - (1/2). Therefore, the particular solution is x = (1/2)e²ᵗ + 17 - (1/2).