Question

Solve the differential equation in Exercises 9–22.19. y²(dy/dx) = 3x²y³ - 6x²

Accepted Answer

Rewrite the given differential equation $$y^{2}$$ \frac{dy}{dx} = $$3x^{2}$$ $$y^{3}$$ - $$6x^{2} to $$isolate $$\frac{dy}{dx}. $$Divide both sides by $$y^{2} ($$assuming $$y 
eq 0) to $$get $$\frac{dy}{dx} = 3x$$^{2}$$ y - \frac{6x$$^{2}$$}{y$$^{2}$$}. $$Recognize that the equation is separable or can be manipulated into a separable form. Try to express all terms involving $$y on $$one side and all terms involving $$x on $$the other side. Rewrite the equation as $$\frac{dy}{dx} = 3x$$^{2}$$ y - 6x$$^{2}$$ y$$^{-2}$$. This suggests grouping terms to separate variables: $$\frac{dy}{dx} = $$x^{2} (3y - 6$$ $$y^{-2}$$)$$. Express the differential equation in differential form: dy$$ = $$x^{2} (3y - 6$$ $$y^{-2}) dx$. Then rearrange to isolate y$ terms with dy$ and x$ terms with dx$: $$\frac{dy}{3y - 6 $$y^{-2}$$} = $$x^{2} dx$. Integrate both sides: integrate $$\int \frac{dy}{3y - 6 $$y^{-2}$$}$$ with respect to y$ and $$\int $$x^{2} dx$ with respect to x$. After integration, include the constant of integration and solve for y$ if possible.

Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²

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Key Concepts

Separable Differential Equations

Algebraic Manipulation of Differential Equations

Integration Techniques