Chapter 9: Differential Equations
Order, Degree & Methods of Solution
1. Order and Degree
| Order | The order of the highest order derivative involved in the equation. |
| Degree | The highest power (exponent) of the highest order derivative. |
| Condition for Degree | Defined only if the equation is a polynomial equation in its derivatives (e.g., no $$\sin(y’), e^{y’}$$, etc.). |
2. General and Particular Solutions
| General Solution | Contains arbitrary constants equal to the Order of the Differential Equation. |
| Particular Solution | Contains no arbitrary constants (obtained by giving specific values to constants). |
3. Method: Homogeneous Differential Equations
Equation where $$dy/dx$$or$$dx/dy$$can be expressed as a function of degree 0.
| Case 1: Form | $$\frac{dy}{dx} = F(\frac{y}{x})$$ |
| Substitution (Case 1) | Put $$y = vx$$and replace$$\frac{dy}{dx}$$with$$v + x\frac{dv}{dx}$$ |
| Case 2: Form | $$\frac{dx}{dy} = F(\frac{x}{y})$$ |
| Substitution (Case 2) | Put $$x = vy$$and replace$$\frac{dx}{dy}$$with$$v + y\frac{dv}{dy}$$ |
4. Method: Linear Differential Equations (LDE)
| Type 1: Standard Form | $$\frac{dy}{dx} + Py = Q$$ (where P, Q are constants or functions of x) |
| Integrating Factor (Type 1) | $$I.F. = e^{\int P dx}$$ |
| Solution Formula (Type 1) | $$y \cdot (I.F.) = \int (Q \cdot I.F.) dx + C$$ |
| Type 2: Standard Form | $$\frac{dx}{dy} + Px = Q$$ (where P, Q are constants or functions of y) |
| Integrating Factor (Type 2) | $$I.F. = e^{\int P dy}$$ |
| Solution Formula (Type 2) | $$x \cdot (I.F.) = \int (Q \cdot I.F.) dy + C$$ |