Question

The general solution of the differential equation 4xy+12x+(2x²+3)y' = 0 is

• A. \(\frac{2x^2+3}{y+3}=C\)
• B. \(\frac{y-3}{2x^2+3}=C\)
• C. \(\frac{y+2}{2x^2+3}=C\)
• D. (y-3)(2x²+3)=C
• E. (y+3)(2x²+3)=C

Answer 1

Answer: (E)

The given differential equation is \( 4xy + 12x + (2x^2 + 3)y' = 0 \).

First, we can rewrite the equation in a more standard form:

\( y' = -\frac{4xy + 12x}{2x^2 + 3} \).

This is a linear first-order differential equation. We can try separating the variables or use an appropriate substitution. After simplifying and solving the equation, we obtain the general solution:

\( (y + 3)(2x^2 + 3) = C \), 

where \( C \) is the constant of integration.

The correct answer is \( (y + 3)(2x^2 + 3) = C \).