Final answer to the problem
Step-by-step Solution
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- Exact Differential Equation
- Linear Differential Equation
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- 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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Rearrange the differential equation
Simplify the product $-(28y+18x+21xy)$
Simplify the product $-(18x+21xy)$
Simplifying
We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=-28$ and $Q(x)=24$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
Compute the integral
The integral of a constant is equal to the constant times the integral's variable
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
So the integrating factor $\mu(x)$ is
Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify
We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$
Integrate both sides of the differential equation with respect to $dx$
Simplify the left side of the differential equation
The integral of a function times a constant ($24$) is equal to the constant times the integral of the function
We can solve the integral $\int e^{-28x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $-28x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{-28}$ out of the integral
Multiply the fraction and term in $24\cdot \left(\frac{1}{-28}\right)\int e^udu$
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $-28x$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiplying fractions $-\frac{6}{7} \times \frac{1}{e^{28x}}$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int24e^{-28x}dx$ and replace the result in the differential equation
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiplying fractions $-\frac{6}{7} \times \frac{1}{e^{\left|-28x\right|}}$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiplying the fraction by $y$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiply the fraction by the term
Any expression multiplied by $1$ is equal to itself
Multiply the fraction by the term
Multiply both sides of the equation by $e^{28x}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
