Calculus: Early Transcendentals, Chapter 9, 9.3, Section 9.3, Problem 16

(dP)/(dt) = sqrt(Pt)
P(1) = 2
To solve, separate the variables.
(dP)/(dt)=sqrtPsqrt t
(dP)/sqrtP=sqrt t dt
P^(-1/2)dP=t^(1/2) dt
Then, take the integral of each side. Apply the formula int x^n dx = x^(n+1)/(n+1)+C .
2P^(1/2) + C = 2/3t^(3/2) + C
2sqrtP+C=(2sqrt t^3)/3+C
2sqrtP+C=(2tsqrtt)/3+C
Since C is a constant, then, we may express the C's in our equation as a single C.
2sqrtP=(2tsqrtt)/3+C
Then, apply the condition P(1)=2 to get the value of C. So, plug-in t=1 and P=2.
2sqrt2 =(2*1sqrt1)/3+C
2sqrt2 =2/3+C
2sqrt2-2/3=C
(6sqrt2-2)/3=C
Then, plug-in the value of C to 2sqrtP=(2tsqrtt)/3+C .
Hence, the implicit solution of the differential equation is
2sqrtP=(2tsqrtt)/3 + (6sqrt2-2)/3
After that, determine the explicit solution. To do so, isolate the P.
2sqrtP=(2tsqrt t + 6sqrt2 - 2)/3
2sqrtP=(2(tsqrtt + 3sqrt2-1))/3
sqrtP=(tsqrtt + 3sqrt2-1)/3
(sqrtP)^2=((tsqrtt + 3sqrt2-1)/3)^2
P=(tsqrtt+3sqrt2-1)^2/9
Therefore, the explicit solution of the given differential equation is P(t)=(tsqrtt+3sqrt2-1)^2/9 .