# How do you solve the differential equation dy/dx=6y^2x, where y(1)=1/25 ?

How do you solve the differential equation $\frac{dy}{dx}=6{y}^{2}x$, where $y\left(1\right)=\frac{1}{25}$ ?
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cisalislewagobc
$\frac{dy}{dx}=6{y}^{2}x$
Separate the variables:
$\frac{1}{{y}^{2}}dy=6xdx$
Integrate both sides:
$\int \frac{1}{{y}^{2}}dy=\int 6xdx$
$-\frac{1}{y}=3{x}^{2}+C$
$\frac{1}{y}=-3{x}^{2}+C$
$y=-\frac{1}{3{x}^{2}+C}$
where C is an arbitrary constant of integration.
Now solve for y(1) to find C:
$y\left(1\right)=\frac{1}{25}=-\frac{1}{3{\left(1\right)}^{2}+C}$
$-\frac{1}{25}=\frac{1}{3+C}$
$3+C=-25$
$C=-28$
Hence, the final solution is:
$y=-\frac{1}{3{x}^{2}-28}$
$⇒y=\frac{1}{28-3{x}^{2}}$