Stewart Section 9.1: Modeling with Differential Equations

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Stewart Section 9.1: Modeling with Differential Equations 0/11
1 Differential Equations - Verification · Level 2
Show that \(y = \dfrac{2}{3} e^x + e^{-2x}\) is a solution of the differential equation \(y' + 2y = 2e^x\).
2 Differential Equations - Verification · Level 2
Verify that \(y = -t \cos t - t\) is a solution of the initial-value problem \(t \dfrac{d y}{d t} = y + t^2 \sin t\), \(\quad y(\pi) = 0\)
3 Differential Equations - Verification · Level 3
(a) For what values of \(r\) does the function \(y = e^{r x}\) satisfy the differential equation \(2y'' + y' - y = 0\)?
(b) If \(r_1\) and \(r_2\) are the values of \(r\) that you found in part (a), show that every member of the family of functions \(y = a e^{r_1 x} + b e^{r_2 x}\) is also a solution.

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4 Differential Equations - Verification · Level 3
(a) For what values of \(k\) does the function \(y = \cos k t\) satisfy the differential equation \(4y'' = -25y\)?
(b) For those values of \(k\), verify that every member of the family of functions \(y = A \sin k t + B \cos k t\) is also a solution.

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5 Differential Equations - Verification · Level 3
Which of the following functions are solutions of the differential equation \(y'' + y = \sin x\)?
(a) \(y = \sin x\)
(b) \(y = \cos x\)
(c) \(y = \dfrac{1}{2} x \sin x\)
(d) \(y = -\dfrac{1}{2} x \cos x\)

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6 Differential Equations - Verification · Level 3
(a) Show that every member of the family of functions \(y = (\ln x + C) / x\) is a solution of the differential equation \(x^2 y' + x y = 1\).
(b) Illustrate part (a) by graphing several members of the family of solutions on a common screen.
(c) Find a solution of the differential equation that satisfies the initial condition \(y(1) = 2\).
(d) Find a solution of the differential equation that satisfies the initial condition \(y(2) = 1\).

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7 Differential Equations - Analysis · Level 3
(a) What can you say about a solution of the equation \(y' = -y^2\) just by looking at the differential equation?
(b) Verify that all members of the family \(y = 1 / (x + C)\) are solutions of the equation in part (a).
(c) Can you think of a solution of the differential equation \(y' = -y^2\) that is not a member of the family in part (b)?
(d) Find a solution of the initial-value problem \(y' = -y^2\), \(\quad y(0) = 0.5\)

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8 Differential Equations - Analysis · Level 3
(a) What can you say about the graph of a solution of the equation \(y' = x y^3\) when \(x\) is close to 0? What if \(x\) is large?
(b) Verify that all members of the family \(y = (c - x^2)^{-\dfrac{1}{2}}\) are solutions of the differential equation \(y' = x y^3\).
(c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)?
(d) Find a solution of the initial-value problem \(y' = x y^3\), \(\quad y(0) = 2\)

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9 Differential Equations - Population Models · Level 3
A population is modeled by the differential equation \( \dfrac{d P}{d t} = 1.2 P \left(1 - \dfrac{P}{4200}\right) \)
(a) For what values of \(P\) is the population increasing?
(b) For what values of \(P\) is the population decreasing?
(c) What are the equilibrium solutions?

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10 Differential Equations - Analysis · Level 4
The Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron \(v(t)\) obeys the differential equation \( \dfrac{d v}{d t} = -v[v^2 - (1 + a)v + a] \) where \(a\) is a positive constant such that \(0 < a < 1\).
(a) For what values of \(v\) is \(v\) unchanging (that is, \(d \dfrac{v}{d} t = 0\))?
(b) For what values of \(v\) is \(v\) increasing?
(c) For what values of \(v\) is \(v\) decreasing?

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11 Differential Equations - Analysis · Level 3
Explain why the functions with the given graphs can't be solutions of the differential equation \( \dfrac{d y}{d t} = e^t (y - 1)^2 \)
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