Question

The integrating factor of the differential equation $2y\frac{dx}{dy}+x=5y^2$ is, $(y\ne 0)$:

• (A) $\sqrt{y}$
• (B) $y^2$
• (C) $y$
• (D) $\frac{1}{\sqrt{y}}$

Answer 1

The Correct Option is A

Explanation:
Given,$2y\frac{dx}{dy}+x=5y^2\dots$ $(i)$Equation $(i)$ can be simplified as,$\frac{dx}{dy}+\frac{x}{2y}=\frac{5}{2}y$On comparing eqn $(i)$ with standard eqn,$\frac{dx}{dy}+Px=Q$,We get$P=\frac{1}{2y}$ and $Q=\frac{5}{2}y$Therefore,$IF=e^{\int Pdy}=e^{\int \frac{1}{2y}dy}$$\Rightarrow IF=e^{\frac{1}{2}\log y}=e^{\log y^{\frac{1}{2}}}$$\Rightarrow IF=\sqrt{y}(\because e^{a\log x}=x^a)$Hence, the correct option is (A).