Expert: Abe Mantell - 1/7/2011

Question
Dy/dx = -2 (y-3)     y(0)=5

Dy/dx = y sqrt(x)   y(0)=1

I am having trouble solving these two problems and finding the answer

Answer
They are both separable.

1. dy/dx=-2(y-3) ==> dy/(y-3)=-2dx, now anti-differentiate both sides
==> ln|y-3|=-2x+C1, now exponentiate both sides
==> y-3=e^(-2x+C1)=e^(-2x)*e^(C1)=e^(-2x)*C
Thus, y(x)=Ce^(-2x)+3
Now for the IC: 5=C(1)+3 ==> C=2 ==> Final Answer: y(x)=2e^(-2x)+3

2. dy/dx=y*sqrt(x), similarly ==> dy/y=x^(1/2)dx ==> ln|y|=(2/3)x^(3/2)+C1
==> y=e^[(2/3)x^(3/2)+C1]=Ce^[(2/3)x^(3/2)]
IC ==> 1=C(1) ==> C=1 ==> y(x)=e^[(2/3)x^(3/2)]

Abe