By A. Harnack

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**Extra resources for An Intro to the Study of the Elements of the Diff and Int Calculus**

**Example text**

The matrix version of the Riccati equation occurs in optimal control. Its practical importance and the fact that it cannot be solved by means of integrals have led to the development of the so-called qualitative theory of diﬀerential equations. 18 Example. 7), is w(t) = 1 3 t + Ct−2 . 17), 3t2 . + 3C The constant C is determined from the IC as C = 2, so the solution of the IVP is y(t) = t + y(t) = t3 t4 + 3t2 + 6t . t3 + 6 Verification with Mathematica R . The input y = (t∧4 + 3 ∗ t∧ 2 + 6 ∗ t)/(t∧3 + 6); Simplify[{D[y,t] + 1 + t∧ 2 - 2 ∗ (t∧ ( - 1) + t) ∗ y + y∧ 2, y /.

Exercises In 1–6, solve the given IVP, or ﬁnd the GS of the DE if no IC is given. 1 4xy + (2x2 − 6y)y = 0, y(1) = 1. 2 12x−4 + 2xy −3 + 6x2 y + (2x3 − 3x2 y −4 )y = 0, y(1) = 2. 3 3x2 y −2 + x−2 y 2 − 2x−3 + (6y −4 − 2x3 y −3 − 2x−1 y)y = 0, y(1) = −1. 4 2x sin y − y cos x + (x2 cos y − sin x)y = 0, y(π/2) = π/2. 5 2e2x − 2 sin(2x) sin y + [2y −3 + cos(2x) cos y]y = 0, y(0) = π/2. 6 (x + y)−1 − x−2 − 4 cos(2x − y) + [(x + y)−1 + 2 cos(2x − y)]y = 0, y(π/2) = 0. In 7–10, ﬁnd an integrating factor (of the indicated form) that makes the DE exact, then solve the given IVP or ﬁnd the GS of the DE if no IC is given.

RC electric circuit. 2 Models with Linear Equations 47 where the positive constants R and C are the resistance and capacitance, respectively, I is the current, and Q is the charge. If V is the voltage of the source, then, since I = Q and since, by Kirchhoﬀ’s law, V = VR + VC , we have V = RI + Q 1 = RQ + Q, C C or, what is the same, Q + 1 1 Q = V. 8) and with an IC of the form Q(0) = Q0 , Q(t) = e 1 R −t/(RC) t eτ /(RC) V (τ ) dτ + Q0 . 8 Example. Suppose that R = 20, C = 1/20, V (t) = cos t, and Q0 = −1/100.