Question

Consider the initial-value problem

y''+yy'=0, y(0)=1, y'(0)= -1

(a) Use the DE and a numerical solver to graph the solution curve.

(b) Find an explicit solution of the IVP. Use a graphing utility to graph this solution.

(c) Find an interval of definition for the solution in part (b).

Short answer

The solution is y= tan[ π /4-x/2]

= - π /2 < (x)<(3π /2)

Step-by-step solution

Solving (a);

Solving the differential equation using graph

y''+yy'=0

y(0)=1, y'(0)= -1

Solving (b):

Let us take u=y'= dy/dx therefore,

y'=u

y''= du/dx

y''= du/dy. dy/dx

y''= du/dy. dy/dx

Therefore we have

y''+yy'=0

u.du/dy+yu=0

du/dy+y=0

du= -ydy

∫ du= -∫ ydy

u=[ (-y²/2)+c₁]

y'=[ (-y²/2)+c₁]

We have y(0)=1 and y’(0)=-1

-1= -1/2+c₁

c₁= -1/2