By N. Finizio
A similar, subtle traditional Differential Equations with glossy purposes through Finizio and Lades is the spine of this article. as well as this are incorporated functions, thoughts and concept of partial distinction equations, distinction equations and Fourier research.
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Extra info for An Introduction to Differential Equations: With Difference Equations, Fourier Series, and Partial Differential Equations
Then the rate of change of the money at time t is given by dt 100 Y. (9) Equation (9) is clearly a separable differential equation. 28. 1 we observed that according to Newton's second law of motion a moving body of mass m and velocity v is governed by the differential equation dt (mv) = kF, (10) where F is the resultant force acting on the body and k is a constant of proportionality. If it happens that F is a function of the velocity v and does not depend explicitly on time, and if m is a constant, then Eq.
Then, from Eq. (2), 1 + w' x' + w'x' (WX)' _ 2 =-+W w w w2 dw = X dx w'x = w2 3w'=InI xI 1 W'X + N, = _ xw x` (separable) +cz;> 3x'=lnI xI + c. Thus, y= EXAMPLE 2 x(3InIxI+c)". Solve the IVP (x2 + y2) dx + 2xy dy = 0 Al) = -I. Solution The differential equation (3) can be written in the form x2 + y2 (5) 2xy which is clearly homogeneous. 2, are satisfied for the IVP (3)-(4). Also, we are looking for a solution through the point (1, -1), and so the division by 2xy in Eq. ] Setting y = wx in Eq. (5), we obtain (wx) x2 + w2x2 2xwx w'x I + w2 i w= 2w I + 3w2 r, w x 2w +3w'dw+dx=0=> 3In(I +3w2)+InIxIc 1 ln[(1 +3w2)I xI']=c=11 +3yx3=c.
Cos x Dividing both sides by 1/cos x (in other words, multiplying both sides by cos x) yields the solution y(x) = (cos x)(c - In cosx), EXAMPLE 2 0