Answer:
To solve the given differential equation \( \frac{dy}{dx} = \frac{6y^2}{\sqrt{x}} \), we can use separation of variables method.
Separating variables:
\[ \frac{1}{y^2} dy = 6 \sqrt{x} dx \]
Integrating both sides:
\[ \int \frac{1}{y^2} dy = \int 6 \sqrt{x} dx \]
\[ -\frac{1}{y} = 6 \cdot \frac{2}{3} x^{\frac{3}{2}} + C \]
\[ -\frac{1}{y} = 4x^{\frac{3}{2}} + C \]
\[ -y = \frac{1}{4x^{\frac{3}{2}} + C} \]
\[ y = -\frac{1}{4x^{\frac{3}{2}} + C} \]
Given the initial condition \( y(1) = \frac{1}{53} \), we can solve for \( C \):
\[ \frac{1}{53} = -\frac{1}{4(1)^{\frac{3}{2}} + C} \]
\[ \frac{1}{53} = -\frac{1}{4 + C} \]
\[ -4 - C = 53 \]
\[ C = -57 \]
So, the particular solution to the differential equation is:
\[ y(x) = -\frac{1}{4x^{\frac{3}{2}} - 57} \]
