Differential Equations 15 Please show work Use the method o

Differential Equations # 15 Please show work

Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. x\' = y, y\' =-x x\' = y, y\' = x x = -2y, y\' = 2x; x(0) = 1, y(0) = 0 x\' = 10y, y\' = -10x; x(0) = 3. y(0) = 4 x\' = 1/2y, y\' =. 8x x\' = 8y, y\' = -2x x\' = y. y\' = 6x - y;x(0) = 1, y(0) = 2 x\' = -y, y\' = 10x - 7y;x(0) = 2, y(O) = -7

Solution

15.

Let, t be the independent variable here

x\'=y/2

y\'=-8x

We differentiate the second equation w.r.t. t and get

y\'\'=-8x\'

Now we substitute the first equation and get

y\'\'=-4y

y\'\'=-2^2y

General solution to this is

y= a sin(2x)+b cos(2x)

Now we substitute this expression of y in second ode and get x

2a cos(2x)-2b sin(2x)=-8x

x=-a/4 cos(2x)+b/4 sin(2x)