First Order Differential Equations Calculator

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 $4\cdot \left(\frac{1}{2}\right)y^2$

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 $C$

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