Solve using separation of variable of y'=x^2,y(0)=2

Solve y'=x^2,y(0)=2

Evaluate

y^{'} = x^{2}, y (0) = 2

Disregard the initial condition

y^{'} = x^{2}

Rewrite the expression

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

Transform the expression

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 1 d y = \int x^{2} d x

Calculate

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y + C_{1} = \int x^{2} d x, C_{1} \in R

Calculate

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y + 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

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

Use the initial condition y (0) = 2 to substitute 0 for x and 2 for y

2 = \frac{0 ^{3}}{3} + C

Calculate

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2 = C

Swap the sides of the equation

C = 2

To find the particular solution, substitute 2 for C in the general solution y = \frac{x ^{3}}{3} + C

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

Solution

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y = \frac{x ^{3} + 6}{3}