Question

Solution of the differential equation (x + 2y³) $\frac{dy}{dx}$= y is –

• A. x = y² (c + y²)
• B. x = y(c – y²)
• C. x = 2y (c – y²)
• D. x = y (c + y²)

Answer 1

Answer: (D)

x = y (c + y²)

Answer 2

We have, (x + 2y³) $\frac{dy}{dx}$ = y Ž y$\frac{dx}{dy}$ = x + 2y³

Ž $\frac{dx}{dy}$ – $\frac{1}{y}$ x = 2y²,

Which is a linear equation, if we take x as the dependent variable.

I.F. = $e^{\int_{}^{}{pdy}}$ = $e^{–\int_{}^{}{\frac{1}{y}dy}}$ = e^{–log y} = $e^{\log\left( \frac{1}{y} \right)}$ = $\frac{1}{y}$.

\ The solution is x. $\frac{1}{y}$ = $\int_{}^{}{2y^{2}}$. $\frac{1}{y}$dy + c

Ž x .$\frac{1}{y}$ = y² + c Ž x = y (c + y²).

Hence (4) is the correct answer.