Differential Equations Tutorial

Differential Equations Tutorial

Order and Degree

  • Order: Highest derivative in the equation
  • Degree: Power of highest derivative (when derivatives are rationalized)
  • Examples:
    • $$5\frac{d^2y}{dx^2} - x\frac{dy}{dx} + (1-x)y = \sin y$$ → Order 2, Degree 1
    • $$5(y'')^2 - y = e^x$$ → Order 2, Degree 2
    • $$\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^3 + y = 0$$ → Order 2, Degree 1
    • $$y' + y^2 x = 2x^3$$ → Order 1, Degree 1

Separable Differential Equations

  • Form: $g(y)dy = f(x)dx$
  • Can rewrite to separate $y$ and $x$ terms
  • Examples:
    • $\frac{dy}{dx} = -\frac{x}{y-3}$ → Separable
    • $\frac{dy}{dx} + x^2y = x$ → Non-separable
    • $y' - 2xy = x^2 - x$ → Non-separable
    • $y' - x^2y^2 = x^2$ → Separable

Solving Separable Equations

  1. Separate variables: $g(y)dy = f(x)dx$
  2. Integrate both sides: $\int g(y)dy = \int f(x)dx$
  3. Add constant of integration $C$

General Solution

  • Contains arbitrary constant $C$
  • Family of curves satisfying the DE

Particular Solution

  • General solution with specific initial condition
  • Use given $(x, y)$ to find $C$

Key Examples

General Solution (from Q3):

  • $\frac{dy}{dx} = x\sqrt{1+x^2}$ → $y = \frac{1}{3}(1+x^2)^{3/2} + C$
  • $e^y\frac{dy}{dx} + \sin x = 0$ → $e^y = \cos x + C$
  • $(x+1)\frac{dy}{dx} = x(y+3)$ → $\ln|y+3| = x - \ln|x+1| + C$
  • $x dy - y dx = 0$ → $y = Cx$

Particular Solution (from Q4):

  • $\frac{dy}{dx} = 1 - \frac{2}{y}$, $y(3)=3$ → $y - 2\ln|y-2| = x - 1$
  • $(xy^2 - xy)dx - 2dy = 0$, $y(0)=2$ → $y = \frac{2}{1 - \frac{x^2}{2}}$