Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- 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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Divide all the terms of the differential equation by $x^3$
Simplify the fraction by $x$
Simplify the fraction $\frac{x^3}{x^3}$ by $x$
Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$
Simplify the fraction by $x$
Subtract the values $3$ and $-2$
Any expression to the power of $1$ is equal to that same expression
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 $P(x)=\frac{3}{x}$ and $Q(x)=\frac{1}{x^{2}}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
Compute the integral
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
Simplify $e^{3\ln\left|x\right|}$ by applying the properties of exponents and logarithms
So the integrating factor $\mu(x)$ is
Multiplying the fraction by $x^3$
Multiplying the fraction by $x^3$
Any expression multiplied by $1$ is equal to itself
Simplify the fraction $\frac{x^3}{x^{2}}$ by $x$
Simplify the fraction $\frac{3yx^3}{x}$ by $x$
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
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 $n=1$
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 $\int xdx$ and replace the result in the differential equation
Multiplying the fraction by $x^2$
Combine all terms into a single fraction with $2$ as common denominator
We can rename $2\cdot C_0$ as other constant
Divide both sides of the equation by $x^3$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
