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Exercise

$\frac{dy}{dx}=x-x^2$

Step-by-step Solution

1

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

$dy=\left(x-x^2\right)dx$
2

Simplify the expression $\left(x-x^2\right)dx$

$dy=x\left(1-x\right)dx$
3

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int1dy=\int x\left(1-x\right)dx$
4

Solve the integral $\int1dy$ and replace the result in the differential equation

$y=\int x\left(1-x\right)dx$
5

Solve the integral $\int x\left(1-x\right)dx$ and replace the result in the differential equation

$y=\frac{1}{2}x^2+\frac{-x^{3}}{3}+C_0$

Final answer to the exercise

$y=\frac{1}{2}x^2+\frac{-x^{3}}{3}+C_0$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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