Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Separable Differential Equation
- Exact Differential Equation
- Linear Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
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We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $\frac{-y}{x}$ from both sides of the equation
Multiplying the fraction by $-1$
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete
Combine and simplify all terms in the same fraction with common denominator $3yx$
We can identify that the differential equation $\frac{dy}{dx}=\frac{x^2+3y^2}{3yx}$ is homogeneous, since it is written in the standard form $\frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and both are homogeneous functions of the same degree
Use the substitution: $y=ux$
Expand and simplify
Integrate both sides of the differential equation, the left side with respect to $u$, and the right side with respect to $x$
Solve the integral $\int3udu$ and replace the result in the differential equation
Solve the integral $\int\frac{1}{x}dx$ and replace the result in the differential equation
Replace $u$ with the value $\frac{y}{x}$
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
Multiplying fractions $\frac{3}{2} \times \frac{y^2}{x^2}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$