Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

532_Differential Equations_[Ch. 9
2781. Find the equation of a curve whose subtangent is equal
to the arithmetic mean of the coordinates of the point of tan-
gency.
2782. Find the equation of a curve for which the segment
cut off on the y-axls by the normal at any point of the curve
is equal to the distance of this point from the origin.
2783*. Find the equation of a curve for which the area contained
between the #-axis, the curve and two ordinates, one of
which is a constant and the other a variable, is equal to the
ratio of the cube of the variable ordinate to the appropriate
abscissa.
2784. Find a curve for which the segment on the y-axis cut
off by any tangent line is equal to the abscissa of the point of
tangency.
Sec. 5. First-Order Linear Differential Equations.
Bernoulli's Equation
1. Linear equations. A differential equation of the form
of degree one in y and y f
is called linear.
If a function Q(jt)=~0, then equation (1)
)-y^Q (x) (1)
takes the form
).y = Q (2)
and is called a homogeneous linear differential equation. In this case,
variables may
the
be separated, and we get the general solution of (2) in
form
the
- P P(X) dx
y = J C-e .
(3)
To solve the inhomogeneous linear equation (1), we apply a method that
is called variation of parameters, which consists in first finding the general
solution of the respective homogeneous linear equation, that is, relationship
(3). Then, assuming here that C is a function of x, we seek the solution
of the inhomogeneous equation (1) in the form of (3). To do this, we put into
(1) y and y' which are found from (3), and then from the differential equation
thus obtained we determine the function C(x). We thus get the general
solution of the inhomogeneous equation (1) in the form
Example I. Solve the equation
= ^/x -fJ
C(x).e
y' tan **/-}- cos x. (4)
Solution. The corresponding homogeneous equation is
Solving it we get:
1 r
^- C
*^I