Final Answer
Step-by-step Solution
Problem to solve:
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Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$
The differential equation $2\left(y+1\right)dy-\left(3x^2+4x+2\right)dx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$
Find the derivative of $M(x,y)$ with respect to $y$
The derivative of the constant function ($-\left(3x^2+4x+2\right)$) is equal to zero
Find the derivative of $N(x,y)$ with respect to $x$
The derivative of the constant function ($2\left(y+1\right)$) is equal to zero
Using the test for exactness, we check that the differential equation is exact
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
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
The integral of a constant is equal to the constant times the integral's variable
The integral of a function times a constant ($3$) is equal to the constant times the integral of the function
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Since $y$ is treated as a constant, we add a function of $y$ as constant of integration
Integrate $M(x,y)$ with respect to $x$ to get
The derivative of the constant function ($-x^{3}-2x^2-2x$) is equal to zero
The derivative of $g(y)$ is $g'(y)$
Now take the partial derivative of $-x^{3}-2x^2-2x$ with respect to $y$ to get
Simplify and isolate $g'(y)$
$x+0=x$, where $x$ is any expression
Rearrange the equation
Solve the product $2\left(y+1\right)$
Set $2\left(y+1\right)$ and $0+g'(y)$ equal to each other and isolate $g'(y)$
Integrate both sides with respect to $y$
Expand the integral $\int\left(2y+2\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a constant is equal to the constant times the integral's variable
The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Find $g(y)$ integrating both sides
We have found our $f(x,y)$ and it equals
Then, the solution to the differential equation is
Group the terms of the equation
Factor the polynomial $y^2+2y$. Add and subtract $\left(\frac{b}{2}\right)^2$, replacing $b$ by it's value $2$
Now, we can factor $y^2+2x+1$ as a squared binomial of the form $\left(x+\frac{b}{2}\right)^2$
We need to isolate the dependent variable $y$, we can do that by subtracting $-1$ from both sides of the equation
We can combine and rename $1+C_0$ as other constant of integration
Removing the variable's exponent
We need to isolate the dependent variable $y$, we can do that by subtracting $1$ from both sides of the equation
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{2x^2+x^{3}+2x+C_0}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
Find the explicit solution to the differential equation