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44 1. First-Order Differential Equations
a) Show that f(t, x)+g(t, x)x ′ = 0 is an exact equation if f x = g t .(This
condition is also implied by exactness.)
b) Use part (a) to check if the following equations are exact. If the equation
is exact, find the general solution by solving H t = f and H x = g
for H. (You may want to review the method of finding potential functions
associated with a conservative force field from your multivariable
calculus course.) Write the solution, or integral curves.
iv) x 3 dt +3tx 2 dx =0.
i) x 3 +3tx 2 x ′ =0.
iii) x ′ sin x−x sin t
= −
t cos x+cos t . v) t(cot x)x ′ = −2.
ii) t 3 + x t +(x2 +lnt)x ′ =0. v) x 2 dt − t 2 dx =0.
17. (Orthogonal trajectories) Solutions to a first-order equation form a set of
integral curves of the form φ(x, y) =C, whereC is a constant. (Here we
are using x and y as variables because the topic is geometry in the plane.)
Theintegralcurves,forexample,maybe the equipotential curves of a conservative
force field, such as an electric field. Every set of integral curves is
defined by a differential equation φ x dx + φ y dy =0,ordy/dx = −φ x /φ y .
We often want to find the integral curves that are perpendicular, or orthogonal,
to the given curves; these are called the orthogonal trajectories.
For example, we may want to find the flux lines of the electric field. Because
orthogonal families of curves are perpendicular, their slopes are negative
reciprocals. Therefore, the integral curves orthogonal to φ(x, y) =C satisfy
the differential equation
dy
dx = φ y
.
φ x
Plot the following integral curves and then find and plot the orthogonal
trajectories.
a) x 2 + y 2 = C. b) xy = C. c) y = Cx 2 .
1.4.2 Applications
Now we consider some practical examples of linear models, which are myriad.
Additional ones are given in the exercises.