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

Sec. 4]_First-Order Homogeneous Differential Equations_331
Solution. Put y = ux', then -f xu' = e u + u or
Q
Integrating, we get w = In In , whence
If
and 6
y x In In .
y
x
2. Equations that reduce to homogeneous equations.
' ! Ue 0, then, putting into equation (2) x w + a, j/ = t;-fp, where
I #2^2 I
the constants a and P are found from the following system of equations,
+ c, = 0,
a 2 a + b$ + c t = 0,
we get a homogeneous differential equation in the variables u and v. If
6 0, then, putting in (2) a,x 4- b^y u, we get an equation with variables
separable.
Integrate the differential equations:
2768. 0' = 1 1.
277 - (x-y)ydx-x*dy = Q.
2769. y^-^.
2771. For the equation (x 2
+y*) dx 2xydy = find the family
of integral curves, and also indicate the curves that pass through
the points (4,0) andj_l,l), respectively.
2772.
2773. xdy ydx = Vx* -\-ifdx.
2
2774. (4x* + 3xy + f/ ) dx + (4y 2
+ 3jvy + jf) dy = 0.
2775. Find the particular solution of the equation (x 1
+ 2xydy = Q, provided that r/=l when x = 2.
Solve the equations:
2776. (2x
1
9777 2/77. ./ - f/
3y*)dx+
2779. Find the equation of a curve that passes through the
point (1,0) and has the property that the segment cut off b\
tangent
the
line on the r/-axis is equal to the radius vector of the
point of tangency.
2780**. What shape should the reflector of a search have so that the rays from a point source of
as a parallel beam?
light
light
are reflected