3 Transform the nonlinear differential equation into a syste

3. Transform the nonlinear differential equation into a system of first order differential equations and determine all equilibrium points of the system.

3. Transform the nonlinear differential equation into a system of first order differential equations 2 y\"(t) + 3y\'\"(t)-y(t)y\'(t) + (y(t)) = 2 and determine all equilibrium points of the system.

Solution

consider the non linear ode

y\'\'\'(t)+3y\'\'(t)-y(t)*y\'(t)+{y(t)}^2=2

to transform in system of first order diff eqn.

let us assume y=x1

y\'=x2

y\'\'=x3

>>>> therefore x1\'=y\'=x2;

x2\'=y\'\'=x3;

x3\'= y\'\' \'= 2-3y\'\'+yy\'-y^2 = 2-3*x3-x1.x2-x1^2.

therefore we get asystem of FO ODE ..

now will find equilibrium point;

we know Equilibrium points are points where the derivative of both x1; x3 and x2 equals zero.

which implies x1\' =0>>>x2 =0

x2\'= 0 implies that x3= 0;

x3\'=0>>>2-3x3+x1.x2-x1^2=0 implies that x1^2 = 2>> x1 = 2, -2

thus equilibrium point are (2,0,0 ) and (2,0,0)