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Exercise

$\frac{dy}{dx}=24+18x+28y+21xy$

Step-by-step Solution

1

Rearrange the differential equation

$\frac{dy}{dx}-\left(28y+18x+21xy\right)=24$
2

Simplifying

$\frac{dy}{dx}-28y-18x-21xy=24$
3

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 $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$
4

To find $\mu(x)$, we first need to calculate $\int P(x)dx$

$\int P(x)dx=\int-28dx=-28x$
5

So the integrating factor $\mu(x)$ is

$\mu(x)=e^{-28x}$
6

Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify

$\frac{dy}{dx}e^{-28x}-28ye^{-28x}-18x-21xy=24e^{-28x}$
7

We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$

$\frac{d}{dx}\left(e^{-28x}y\right)=24e^{-28x}$
8

Integrate both sides of the differential equation with respect to $dx$

$\int\frac{d}{dx}\left(e^{-28x}y\right)dx=\int24e^{-28x}dx$
9

Simplify the left side of the differential equation

$e^{-28x}y=\int24e^{-28x}dx$
10

Solve the integral $\int24e^{-28x}dx$ and replace the result in the differential equation

$e^{-28x}y=\frac{-6}{7e^{28x}}+C_0$
11

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{e^{28x}}y=\frac{-6}{7e^{28x}}+C_0$
12

Multiply the fraction by the term

$\frac{y}{e^{28x}}=\frac{-6}{7e^{28x}}+C_0$
13

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

$y=\left(\frac{-6}{7e^{28x}}+C_0\right)e^{28x}$

Final answer to the exercise

$y=\left(\frac{-6}{7e^{28x}}+C_0\right)e^{28x}$

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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