Find the particular solution that satisfies the initial condition

Question

sqrt(x) + sqrt(yy’) = 0

y(1)=4

Answer 1

We can use separation of variables.

√x + √yy' = 0 ⇔

x + yy' = 0 [square both sides] ⇔

y dy/dx = -x ⇔

y dy = - x dx [all the y terms including dy can be moved to one side and vice verse with x].

Now we can integrate both sides.

∫ydy = ∫-xdx ⇔

1/2y² = -1/2x² + C₁ ⇔

y= √(-x + 2C₁)

Now to satisfy the I.C.

y(1)=4 ⇒ 4=√(-1 + 2C₁)⇒ C₁= 17/2

y= √(-x+2(17/2)) ⇔ y=√(-x+17)