Final answer:
The differential equation y' - 3x² y = 0 is a first-order linear ordinary differential equation. By using an integrating factor, we can find the general solution to be y = Ce^{x³}, where C is any real number.
Step-by-step explanation:
We are asked to solve the differential equation y' - 3x² y = 0. This type of differential equation is known as a first-order linear ordinary differential equation.
To solve it, we can use the method of separation of variables or an integrating factor. Since the equation is already in the form of a first-order linear equation, we will use an integrating factor. The integrating factor, μ(x), can be found by using the formula μ(x) = e^{∫ P(x) dx}, where P(x) is the coefficient of y in the differential equation.
Thus, for our equation:
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- P(x) = -3x², so μ(x) = e^{∫ (-3x²) dx} = e^{-x³}.
Next, we multiply both sides of our differential equation by this integrating factor:
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- e^{-x³}(y' -3 x² y) = 0.
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- This simplifies to (e^{-x³}y)' = 0.
We then integrate both sides with respect to x:
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- ∫ (e^{-x³}y)' dx = ∫ 0 dx,
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- e^{-x³}y = C,
where C is the constant of integration.
Finally, solve for y by multiplying both sides by e^{x³}:
where C is any real number. This is the general solution to the differential equation.