Separation of Variables Math Example 3

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

Example 3

easy
Solve dy/dx=x/ydy/dx = x/y with y(0)=3y(0)=3.

Solution

  1. 1
    y dy=x dxy\,dy = x\,dx. Integrate: y2/2=x2/2+Cy^2/2 = x^2/2+C.
  2. 2
    y(0)=3y(0)=3: C=9/2C=9/2. y=x2+9y = \sqrt{x^2+9}.

Answer

y=x2+9y = \sqrt{x^2+9}
Separate, integrate, apply IC, take positive root.

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.

Learn more about Separation of Variables →

More Separation of Variables Examples

Example 1 easy

Solve [formula] with [formula].

Example 2 hard

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

Example 4 medium

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

← All Separation of Variables Examples Separation of Variables Concept