Solving Differential Equations by Separation of Variables

Example Problem 1:

Given: ( \frac{dy}{dx} = \frac{x^2}{y^2} )

• Separate Variables:

  • Cross multiply:
    • ( y^2 dy = x^2 dx )

• Integrate Both Sides:

  • Left Side: ( \int y^2 dy = \frac{y^3}{3} )
  • Right Side: ( \int x^2 dx = \frac{x^3}{3} + C )

• Rearranging:

  • Multiply by 3: ( y^3 = x^3 + 3C )
  • Let ( C' = 3C ): ( y^3 = x^3 + C' )
  • Simplify: ( y = \sqrt[3]{x^3 + C} )

• Particular Solution (given ( y(1) = 2 )):

  • Set up equation: ( 2^3 = 1^3 + C )
  • Solve for ( C ):
    • ( 8 = 1 + C )
    • ( C = 7 )
  • Particular Solution: ( y = \sqrt[3]{x^3 + 7} )

Example Problem 2:

Given: ( y' = xy )

• Separate Variables:

  • Rewrite: ( dy = xy dx )
  • Divide by ( y ): ( \frac{1}{y} dy = x dx )

• Integrate Both Sides:

  • Left Side: ( \int \frac{1}{y} dy = \ln |y| )
  • Right Side: ( \int x dx = \frac{x^2}{2} + C )

• Exponentiate to Solve for y:

  • ( |y| = e^{\frac{x^2}{2} + C} )
  • Simplify: ( y = Ce^{\frac{x^2}{2}} )

• Particular Solution (given ( y(0) = 5 )):

  • Set up equation: ( 5 = Ce^{0} )
  • Solve for ( C ): ( C = 5 )
  • Particular Solution: ( y = 5e^{\frac{x^2}{2}} )

Example Problem 3:

Given: ( \frac{dy}{dx} = y^2 + 1 )

• Separate Variables:

  • Multiply by ( dx ): ( dy = (y^2 + 1) dx )
  • Rearrange: ( \frac{1}{y^2 + 1} dy = dx )

• Integrate Both Sides:

  • Left Side: ( \int \frac{1}{y^2 + 1} dy = \tan^{-1}(y) )
  • Right Side: ( \int dx = x + C )

• Solve for y:

  • Take tangent of both sides: ( y = \tan(x + C) )

• Particular Solution (given ( y(1) = 0 )):

  • Set up equation: ( 0 = \tan(1 + C) )
  • Solve for ( C ): ( 1 + C = n\pi \Rightarrow C = n\pi - 1 )
  • Specific Solution: ( y = \tan(x + C) ) for any integer ( n )

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Note: Always check your initial conditions and solutions for correctness.