Solve using separation of variable of \frac{dy}{dx} + 3 x^{2} y = x^{2}

y=\frac{C+e^{\left(x^{3}\right)}}{3e^{\left(x^{3}\right)}},C \in \mathbb{R}

Question :

fracdydx + 3 x^2 y = x^2

y = \frac{C + e ^{(x^{3})}}{3 e ^{(x^{3})}}, C \in R

Evaluate

\frac{d y}{d x} + 3 x^{2} y = x^{2}

Move the expression to the right side

\frac{d y}{d x} = x^{2} - 3 x^{2} y

Rewrite the expression

\frac{d y}{d x} = x^{2} (1 - 3 y)

Rewrite the expression

\frac{1}{1 - 3 y} \times \frac{d y}{d x} = x^{2} (1 - 3 y) \times \frac{1}{1 - 3 y}

Multiply the terms

\frac{1}{1 - 3 y} \times \frac{d y}{d x} = x^{2}

Transform the expression

\frac{1}{1 - 3 y} \times d y = x^{2} d x

Integrate the left-hand side of the equation with respect to y and the right-hand side of the equation with respect to x

\int \frac{1}{1 - 3 y} d y = \int x^{2} d x

Calculate

More Steps

- \frac{1}{3} ln (y - \frac{1}{3}) + C_{1} = \int x^{2} d x, C_{1} \in R

Calculate

More Steps

- \frac{1}{3} ln (y - \frac{1}{3}) + C_{1} = \frac{x ^{3}}{3} + C_{2}, C_{1} \in R, C_{2} \in R

Since the integral constants C_{1} and C_{2} are arbitrary constants, replace them with constant C

- \frac{1}{3} ln (y - \frac{1}{3}) = \frac{x ^{3}}{3} + C, C \in R

Calculate

More Steps

y = \frac{3 e ^{- x^{3} + C} + 1}{3}, C \in R

Rewrite the expression

More Steps

y = \frac{3 C e ^{- x^{3}} + 1}{3}, C \in R

Solution

y = \frac{C + e ^{(x^{3})}}{3 e ^{(x^{3})}}, C \in R

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