Find a general solution of the system of equations dxdt x2y

Find a general solution of the system of equations:

dx/dt =x2y

dy/dt =2x+y

Solution

From first equation

y=(x-x\')/2

y\'=(x\'-x\'\')/2=2x+y=2x+(x-x\')/2

(x\'-x\'\')/2=2x+(x-x\')/2

x\'-x\'\'=4x+x-x\'

x\'\'-2x\'+5x=0

We have a linear homogeneous ode with constant coefficients so solution is of the form:x=e^{kt}

Substituting gives

k^2-2k+5=0

Solving gives:

k=1+2i,k=1-2i

So,

x=e^t(A sin(2t)+B cos(2t))

y=(x-x\')/2

x\'=x+e^t(2A cos(2t)-2B sin(2t))

(x-x\')/2=e^t(A cos(2t)-B sin(2t))

y=e^t(A cos(2t)-B sin(2t))