Please help me I do not have any idea Consider the following

Please help me

I do not have any idea

Consider the following IVP: y\" + yy\' = 0, y(0) = 1, y\'(0) = -1 Use the dsolve command in MATLAB to find an explicit solution of the IVP. Using MATLAB, plot your solution from part a) over -5pi lessthanorequalto x lessthanorequalto 5pi. Give an interval of definition for your solution from part a).|

Solution

y\"+y\'=0

Let u = dy = dy/dt,

then,

y\'\' = du/dt = du/dy * dy/dt = u*du/dy

. u*du/dy = y*u = 0
du/dy = -y

This is a separable equation:

du = -y dy

u = c - (y^2)/2   where c is the constant of integration.

Back substitute for u:

dy/dt = c - (y^2)/2

dy/dt = (1/2)*(C - y^2)

where C = 2c is just another way of writing the constant.

This equation is also separable:

dy/(C - y^2) = dt/2

(1/sqrt(C))*arctanh(y/sqrt(C)) = t/2 + d

where d is the second constant of integration.

arctanh(y/sqrt(C)) = sqrt(C)*t/2 + d*sqrt(C)

y(t) = sqrt(C)*tanh(sqrt(C)*t/2 + d*sqrt(C))

y(t) = sqrt(C)*tanh(sqrt(C)*t/2 + d*sqrt(C))

We can make this simpler looking by setting a = sqrt(C) and b = d*sqrt(C):

y(t) = a*tanh(a*t/2 + b)