Solve the differential equation. (dy)/(dx)=6sqrt(xy)

Recognize type of differential equation: Recognize the type of differential equation. This is a first-order differential equation that can be solved by separation of variables, where we separate the x terms and y terms on different sides of the equation.Separate the variables: Separate the variables. We want to get all y terms on one side and all x terms on the other side. To do this, we divide both sides by sqrt(y) and multiply both sides by dx to get: (dy)/(sqrt(y)) = 6sqrt(x)dxIntegrate both sides: Integrate both sides. We integrate the left side with respect to y and the right side with respect to x: ∫(1/sqrt(y))dy = ∫6sqrt(x)dxPerform the integration: Perform the integration. The integral of 1/sqrt(y) with respect to y is 2sqrt(y), and the integral of 6sqrt(x) with respect to x is 4x^(3/2), so we have: 2sqrt(y) = 4x^(3/2) + C, where C is the constant of integration.Solve for y: Solve for y. To solve for y, we square both sides to get rid of the square root: (2sqrt(y))^2 = (4x^(3/2) + C)^2 y = (2x^(3/2) + C/2)^2Check for math errors: Check for math errors. We need to ensure that the squaring process was done correctly. Squaring the right side should distribute the square to both terms and the cross term: y = 4x^3 + 2*C*x^(3/2) + (C/2)^2

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Solve the differential equation. $\newline$ $\frac{dy}{dx}=6\sqrt{xy}$

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Q. Solve the differential equation.

\newline

\frac{dy}{dx}=6\sqrt{xy}

Recognize type of differential equation: Recognize the type of differential equation. $\newline$ This is a first-order differential equation that can be solved by separation of variables, where we separate the $x$ terms and $y$ terms on different sides of the equation.
Separate the variables: Separate the variables. $\newline$ We want to get all $y$ terms on one side and all $x$ terms on the other side. To do this, we divide both sides by $\sqrt{y}$ and multiply both sides by $dx$ to get: $\newline$ $\frac{dy}{\sqrt{y}} = 6\sqrt{x}dx$
Integrate both sides: Integrate both sides. $\newline$ We integrate the left side with respect to $y$ and the right side with respect to $x$ : $\newline$ $\int(1/\sqrt{y})\,dy = \int 6\sqrt{x}\,dx$
Perform the integration: Perform the integration. $\newline$ The integral of $\frac{1}{\sqrt{y}}$ with respect to $y$ is $2\sqrt{y}$ , and the integral of $6\sqrt{x}$ with respect to $x$ is $4x^{\frac{3}{2}}$ , so we have: $\newline$ $2\sqrt{y} = 4x^{\frac{3}{2}} + C$ , where $C$ is the constant of integration.
Solve for y: Solve for y. $\newline$ To solve for y, we square both sides to get rid of the square root: $\newline$ $(2\sqrt{y})^2 = (4x^{\frac{3}{2}} + C)^2$ $\newline$ y = $(2x^{\frac{3}{2}} + \frac{C}{2})^2$
Check for math errors: Check for math errors. $\newline$ We need to ensure that the squaring process was done correctly. Squaring the right side should distribute the square to both terms and the cross term: $\newline$ $y = 4x^3 + 2\cdot C\cdot x^{\frac{3}{2}} + \left(\frac{C}{2}\right)^2$