Answer:
Step-by-step explanation:
To solve the differential equation
dy/dx = 5x^4(1 + y²)^(3/2)
First, separate the variables
dy/(1 + y²)^(3/2) = 5x^4 dx
Now, integrate both sides
To integrate dy/(1 + y²)^(3/2), use the substitution y = tan(u)
dy = (1/cos²u)du
So,
dy/(1 + y²)^(3/2) = [(1/cos²u)/(1 + tan²u)^(3/2)]du
= (1/cos²u)/(1 + (sin²u/cos²u))^(3/2)
Because cos²u + sin²u = 1 (Trigonometric identity),
The equation becomes
[1/(1/cos²u)^(3/2) × 1/cos²u] du
= cos³u/cos²u
= cosu
Integral of cosu = sinu
But y = tanu
Therefore u = arctany
We then have
cos(arctany) = y/√(1 + y²)
Now, the integral of the equation
dy/(1 + y²)^(3/2) = 5x^4 dx
Is
y/√(1 + y²) = x^5 + C
So
y - (x^5 + C)√(1 + y²) = 0
is the required implicit solution
is a factor of a polynomial, then 
is a root of this polynomial.
