Separation of Variables Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Solve dy/dx=xydy/dx = xy with y(0)=2y(0)=2.

Solution

  1. 1
    Separate: dy/y=x dxdy/y = x\,dx. Integrate: ln⁡∣y∣=x2/2+C\ln|y| = x^2/2+C.
  2. 2
    y=Aex2/2y = Ae^{x^2/2}. y(0)=2⇒A=2y(0)=2 \Rightarrow A=2.
  3. 3
    Solution: y=2ex2/2y = 2e^{x^2/2}.

Answer

y=2ex2/2y = 2e^{x^2/2}
Separate, integrate both sides, exponentiate, apply IC.

About Separation of Variables

A method for solving first-order DEs of the form dydx=f(x)⋅g(y)\frac{dy}{dx} = f(x) \cdot g(y): rearrange to dyg(y)=f(x) dx\frac{dy}{g(y)} = f(x)\,dx, then integrate both sides.

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More Separation of Variables Examples

Example 2 hard

Solve the logistic DE [formula] with [formula].

Example 3 easy

Solve [formula] with [formula].

Example 4 medium

Solve Newton's cooling: [formula], [formula].

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