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

Sec. 15]_Systems of Differential Equations_359
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3077. Find the particular solution of the equation
*V xy' + y = 2x
that satisfies the initial conditions y = 0, */' = ! when *=1.
Sec. 15. Systems of Differential Equations
Method of elimination. To find the solution, for instance, of a normal
system of two first-order differential equations, that is, of a system of the
form
solved for the derivatives of the desired functions, we differentiate one of
them with respect to x. We have, for example,
Determining z from the first equation of the system (1) and substituting the
value found,
/ A.. \
into equation (2), we get a second-order equation with one unknown function
u. Solving it, we find
where C, and C 2 are arbitrary constants. Substituting function (4) into formula
(3), we determine the function z without new integrations. The set of
formulas (3) and (4), where y is replaced by \|>, yields the general solution
of the system (1).
Example. Solve the system
z , 3 f
+'-'-T*
Solution. We differentiate the first equation with respect to x:
^ dx*^ + 2^ dx^ + 4^-4. dx
\ / dy \
From the first equation we determine ^ = -T- ( l+4x ~ 2y j and then
from the second we will have -^ = -5- ** + * + -; -75- y TT"- Put ting z
ax & 4 & T ax
and j- into the equation obtained after differentiation, we arrive at a second-
order equation in one unknown y:
(3)
(4)