Here, we show you a step-by-step solved example of first order differential equations. This solution was automatically generated by our smart calculator:
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 ($4$) 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 $4\cdot \left(\frac{1}{2}\right)y^2$
Solve the integral $\int 4ydy$ and replace the result in the differential equation
The integral of a function times a constant ($5$) 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 $5\left(\frac{x^{3}}{3}\right)$
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 $\int 5x^2dx$ and replace the result in the differential equation
Multiplying the fraction by $x^{3}$
Combine all terms into a single fraction with $3$ as common denominator
We can rename $3\cdot C_0$ as other constant
Divide both sides of the equation by $2$
Removing the variable's exponent
Cancel exponents $2$ and $1$
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{\frac{5x^{3}+C_1}{6}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
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 the fraction by $-1$
Combining all solutions, the $2$ solutions of the equation are
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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