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Exercise

$\frac{dy}{dx}=\frac{3x^2+4x+2}{2\left(y+1\right)}$

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

$2\left(y+1\right)dy=\left(3x^2+4x+2\right)dx$
2

Simplify the expression $2\left(y+1\right)dy$

$\left(2y+2\right)dy=\left(3x^2+4x+2\right)dx$
3

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

$\int\left(2y+2\right)dy=\int\left(3x^2+4x+2\right)dx$
4

Expand the integral $\int\left(2y+2\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2ydy+\int2dy=\int\left(3x^2+4x+2\right)dx$
5

Expand the integral $\int\left(3x^2+4x+2\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2ydy+\int2dy=\int3x^2dx+\int4xdx+\int2dx$
6

Solve the integral $\int2ydy+\int2dy$ and replace the result in the differential equation

$y^2+2y=\int3x^2dx+\int4xdx+\int2dx$
7

Solve the integral $\int3x^2dx+\int4xdx+\int2dx$ and replace the result in the differential equation

$y^2+2y=x^{3}+2x^2+2x+C_0$
8

Find the explicit solution to the differential equation. We need to isolate the variable $y$

$y=-1+\sqrt{x^{3}+2x^2+2x+C_0+1},\:y=-1-\sqrt{x^{3}+2x^2+2x+C_0+1}$

Final answer to the exercise

$y=-1+\sqrt{x^{3}+2x^2+2x+C_0+1},\:y=-1-\sqrt{x^{3}+2x^2+2x+C_0+1}$

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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