Final answer:
To solve the equation dy/dt = y+2∆(t-3), separate the variables, integrate, solve for y, and determine the particular solution using the initial condition y(0). The general solution is y = e^C * e^(∆t), and the particular solution is y = e^C.
Step-by-step explanation:
To solve the equation dy/dt = y+2∆(t-3), we can start by separating the variables. Move the y term to one side and the t term to the other side. The equation becomes dy/(y+2) = ∆ dt. Then, integrate both sides. The integral of dy/(y+2) is ln|y+2| and the integral of ∆ dt is just ∆t.
Next, we can solve for y. Taking the natural logarithm of both sides, we get ln|y+2| = ∆t + C, where C is the constant of integration. Finally, exponentiate both sides to eliminate the logarithm. The solution is y+2 = e^(∆t + C), which can be simplified to y = e^C * e^(∆t). This is the general solution of the equation.
To find the particular solution, we need to use the initial condition y(0). Plugging in t=0, we get y = e^C * e^(∆*0). Since e^0 = 1, the equation simplifies to y = e^C. Therefore, the particular solution is y = e^C.