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Answer :
Final answer:
To solve the given differential equation, rewrite it in terms of dy/dx and separate the variables y and x. Integrate both sides with respect to x and solve for y.
Explanation:
To solve the differential equation 3x² y' = 45x⁴ + y³, we can start by rewriting the equation in the form dy/dx = (45x⁴ + y³)/3x². We can rearrange this equation to separate the variables y and x.
Next, we can integrate both sides of the equation with respect to x to find the solution for y.
By integrating, we get ∫1/y³ dy = ∫15x² + C dx.
Solving these integrals, we get -1/(2y²) = 5x³/3 + Cx + D.
Rearranging the equation and simplifying, we have y² = -2/(5x³/3 + Cx + D).
Therefore, the solution to the differential equation is y = ±√(-2/(5x³/3 + Cx + D)), where C and D are constants determined by initial conditions or given values.
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