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Calculus/differential equations and separation of variables

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Expert: - 3/27/2009


Question
If dy/dx = x^3 * 1/y(x) and given y(0) = 1, how do you find y(x)?

Answer
Multiply both sides by y(x) and dx giving y(x) dy = x^3 dx.

Make y(x) just y, so you have y dy = x^3 dx.

Integrate both sides, giving y^2/2 = x^4/4 + C.

Multiply both sides by 2, giving y^2 = x^4/2 + 2C,
and note the 2C is still some constant C, so you have
 y^2 = x^4/2 + C.

Take the squareroot of both sides and get
y = √(x^4/2 + C) or y = -√(x^4/2 + C).  

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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