How do you solve the differential equation dydx=6y2x, where y(1)=125?

Question: "How do you solve the differential equation dydx=6y2x,..."

Magdalena Warren

2022-04-09

How do you solve the differential equation

\frac{d y}{d x} = 6 y^{2} x

, where

y (1) = \frac{1}{25}

Coupewopmergorlpt

Beginner2022-04-10Added 13 answers

We are given:

\frac{d y}{d x} = 6 y^{2} x

Separate the variables:

\frac{1}{y^{2}} d y = 6 x d x

Integrate both sides:

\int \frac{1}{y^{2}} d y = \int 6 x d x

- \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 (1) = \frac{1}{25} = - \frac{1}{3 {(1)}^{2} + C}

- \frac{1}{25} = \frac{1}{3 + 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}

\Rightarrow y = \frac{1}{28 - 3 x^{2}}

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