The University of Texas at Dallas Differential Equation Problems hi, need help on this entry level differential equation assignment, need immediately, woul

The University of Texas at Dallas Differential Equation Problems hi, need help on this entry level differential equation assignment, need immediately, would like you to show step by step work and explain how to solve them 1. (15 pts.)
a) Verify ( do not find it … !!!) that the given function is a solution of the differential equation.
vý – 24y = 1; y=+* [*e**dste
b) Determine the values of r for which the given differential equation has a solution of the form y=e”.
y” +y’ – by = 0
c) Determine the long term behavior of the solution of the given directional field. Draw over the plot, some
solutions.
0
0
0.51
1.5
2.5
3
3.5
4
2
t
2. (10 pts) Solve the Initial Value Problems (IVP’s)
a) y
2.2
y+ray
y(0) = -2.
sin(t).
) tu + 2y = ; y(-/2) = a; 0 is the decay
rate.
(a) If 100 mg of thorium-234 decays to 82.04 mg in 1 week, determine the decay rate r.
(b) Find an expression for the amount, Q(t) of thorium-234 present at any time t.
(c) Find the time required for the thorium-234 to decay to one-half its original amount.
6. ( 15 pts. ) Consider an electric circuit containing a capacitor, resistor, and battery; The charge Q(t) on the
capacitor satisfies the equation
=V
dt
(This equation results from Kirchhoff’s laws )* where R is the resistance, C is the capacitance, and V is the
constant voltage supplied by the battery.
(a) If Q(0) = 0, find Q(t) at any time t, and sketch the graph of Q versus t.
(b) Find the limiting value QL that Q(t) approaches after a long time.
(c) Suppose that Qltı) = QL and that at time t = ti the battery is removed and the circuit is closed again.
Find Q(t) for t > t, and sketch its graph.
* Gustav Robert Kirchhoff (12 March 1824 – 17 October 1887) was a German physicist who contributed
to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by
heated objects.

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