Final answer to the problem
Step-by-step Solution
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- 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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Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
The integral of a function times a constant ($3$) 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 $2$
Simplify the fraction $3\left(\frac{y^{3}}{3}\right)$
Solve the integral $\int3y^2dy$ and replace the result in the differential equation
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 $n=1$
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)x^2$
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 $\int2xdx$ and replace the result in the differential equation
Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{3}$
Cancel exponents $3$ and $1$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
