How Do You Plot Solutions and Understand Coefficients in Differential Equations?

[dillonmhudson]

Homework Statement

Solve each of the following initial value problems and plot solutions for several values of y_0.
a) dy/dt=-y+5, y(0)=y_0
b) dy/dt=-2y+5, y(0)=y_0
c) dy/dt=-2y+10, y(0)=y_0

Homework Equations

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The Attempt at a Solution

I have solved all of the equations for general solutions but am having trouble plotting them (these are correct in the back of the book)
a) y=5+(y_0-5)*e^t
b) y=5/2+(y_0-5/2)*e^2t
c) y=5+(y_0-5)*e^2t

TIA

 

[dillonmhudson]

1. Homework Statement
Use method of undetermined coefficients to find the general solution of the given equation:
y'+2y=2t+3sin(t)
y=ce^-2t+Y(t) (where Y(t) is picked)
Solutions manual says: Y(t)=At+B+Ccost(t)-Dsin(t)

so that,
y=ce^-2t+At+B+Ccos(t)-Dsin(t)
I understand the "At+B" but why is it "Ccos(t)-Dsin(t)" and why is there a minus?

Thanks again