Week 4

Chapters (Differential Equations For Engineers)

• 2.4 Linear First-Order Equations
  • 2.4.2 Bernoulli Differential Equations

Videos:

• Bernoulli Differential Equations
  • Professor Leonard: How to Solve Bernoulli Differential Equations
• Ricatti Differential Equations
  • SkanCity Academy: Ricatti Differential Equation 1

Ricatti Differential Equations:

Ricatti Equations come in the form of $$ \frac{dy}{dx} + P(x) = R(x)y^2 + Q(x) $$ Because you are still learning the fundementals, you will be given a hint to solve Ricatti, a particular solution. Using it, you will be able to convert the Ricatti D.E to a Linear D.E.

Example, use the particular solution \( y_1 = \frac{-1}{x} \) to find the general solution of \( x^2(\frac{dy}{dx}+y^2) = 2 \)

Solution

Use the formula \( y = y_1 + \frac{1}{u} \) to convert the Ricatti D.E to a Linear D.E. Then, differentiate \( y \) with respect to \( x \) $$ y = \frac{-1}{x} + \frac{1}{u} \qquad \frac{dy}{dx} = \frac{1}{x^2} + \frac{-1}{u^2} \cdot \frac{du}{dx} $$ Then solve $$ x^2 \left[ \Big(\frac{1}{x^2} + \frac{-1}{u^2} \cdot \frac{du}{dx} \Big) + \Big( \frac{-1}{x} + \frac{1}{u} \Big)^2 \right] = 2 $$ After simplifying, you'll get $$ \frac{du}{dx} + \frac{2}{x}u = 1 $$ Which is a Linear Differential Equation that you know how to solve.