Question

Find the general solution of the differential equation \(\frac{dy}{dx}=5x^2+2\).

(a) 10x^3+12x-3y^2+C=0

(b) 12x-3y^2+C=0

(c) 10x^3+12x-y^2+C=0

(d) 10x^2-3y^2+C=0

I got this question in class test.

My question comes from Methods of Solving First Order & First Degree Differential Equations topic in section Differential Equations of Mathematics – Class 12

Answer 1

Correct choice is (a) 10x^3+12x-3y^2+C=0

To explain I would say: Given that, \(\frac{dy}{dx}=5x^2+2\)

Separating the variables, we get

dy=(5x^2+2)dx –(1)

Integrating both sides of (1), we get

\(\int y \,dy=\int 5x^2+2 \,dx\)

\(\frac{y^2}{2}=\frac{5x^3}{3}+2x+C_1\)

3y^2=\(10x^3+12x+6C_1\)

10x^3+12x-3y^2+C=0 (where 6C1=C)