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Sec. 3] Differential Equations with Variables Separable 327
Sec. 3. First-Order Differential Equations with Variables Separable.
Orthogonal Trajectories
1. First-order equations with variables separable. An equation with variables
separable is a first-order equation of the type
y' = f(x)g(y} (i)
X (x) Y (y) dx + X, (x) Y, (y) dy = Q (!')
Dividing both sides of equation (1) by g(y) and multiplying by dx, we get
- = f(x)dx Whence, by integrating, we get the general integral of equa-
tion (1) in the form
Similarly, dividing both sides of equation (!') by X, (x) Y (y) and integrating,
we get the general integral of (!') in the form
If for some value y = yQ we have (r/ )=0, then the function y = ti Q
is
also (as is directly evident) a solution of equation (1) Similarly, the straight
lines x a and y-b will be the integral curves of equation (!'), if a and b
are, respectively, the roots of the equations X, (*)() and Y = (*/) 0, by the
ieft sides of which we had to divide the initial equation.
Example 1. Solve the equation
In particular, find the solution that satisfies the initial conditions
Solution. Equation (3) may
be written in the torm
dx~~ x
Whence, separating variables, we have
and, consequently,
In |
y
= In | | x\ + ln C |t
where the arbitrary constant In C, is taken in logarithmic form. After taking
antilogarithms we get the general solution
where C= C,.
When dividing dividi by y we could lose the solution =0. but the latter is
' ' -
contained in the formula ila (4) for C = 0.
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