Answer to Question #91479 in Differential Equations for Sajid

Question

Q.Choose the correct answer.
Q. The initial value value problem dy/dx=|y|1/2 ,y(0)=0 has:
a) unique solution
b) No solution
c) Infinitely many solution
d) Two solutions.

Answer 1

a) The initial value problem dy/dx=|y|1/2, y(0)=0 has unique solution.

Explanation.

Let us raise the each side of equation in square

(y')²=y²1/4; <=> (y'-y/2)(y'+y/2)=0;

1) y'-y/2=0; dy/y=1/2; y=Cexp(x/2);

Hence,

y'=Cexp(x/2)/2;

Let us substitute it in the initial equation

Cexp(x/2)/2=|Cexp(x/2)/2|; C=|C|>=0.

y(0)=Cexp(0)=C; y(0)=0; <=> C=0; thus y=0exp(x/2)=0.

2) y'+y/2=0; dy/y=-1/2; y=Cexp(-x/2);

Hence,

y'=-Cexp(x/2)/2;

Let us substitute it in the initial equation

-Cexp(x/2)/2=|-Cexp(x/2)/2|; -C=|C|>=0; C<=0.

y(0)=Cexp(-0)=C; y(0)=0; <=> C=0; thus y=0exp(x/2)=0.

Hence, the initial value problem dy/dx=|y|1/2, y(0)=0 has unique solution y=0.