04 Linear DE
Linear DE General Form¶
\[ y' + P y = Q \]
where \(y\) is the dependent variable
Solution¶
-
Find IF \(= e^{\int P(x) dx}\)
-
Find general solution
\[ y \times IF = \int \Big( Q \times IF \Big) dx \quad + c \]
Bernoulli’s DE¶
\[ y' + P y = Q y^n \]
Solution¶
- Divide both sides by \(y^n\)
\[ y^{-n} y' + P y^{1-n} = Q \]
- Take \(z = y^{1-n}\)
\[ \begin{aligned} z' &= (1-n) y^{(1-n)-1} y' \\ y^{-n} y' &= \left( \frac{1}{1-n} \right) z' \end{aligned} \]
- Convert into a Linear DE
\[ \begin{aligned} \left( \frac{1}{1-n} \right) z' + P z &= Q \\ z' + \underbrace{(1-n) P}_{P\text{ of linear DE}} \ z &= (1-n) Q \end{aligned} \]
- Solving using Linear DE method in terms of \(z\)
\[ \begin{aligned} \text{IF} &= (1-n) \int P dx \\ z \times \text{IF} &= (1-n) \int (Q \times \text{IF}) dx \quad + c \end{aligned} \]
- Put \(z = y^{1-n}\) back into this