Chapter 2 Solution of Differential Equations Separation of Variables Dr. Khawaja Zafar Elahi
Chapter 2
Chapter 2
Example Chapter 2
Chapter 2
Chapter 2
Domain Chapter 2
Partial derviavtive with respect to y Chapter 2
EXAMPLE STEP 1 STEP 2 STEP 3 Chapter 2
Solution STEP 4 STEP 5 STEP 6 Chapter 2
STEP 1 STEP 2 STEP 3 STEP 4 Chapter 2
Chapter 2
Method for solving First Order Differential Equations Chapter 2
Homogeneous differential Equations Exact Differential Equations Methods for solving First order Differential Equations Separable Variable Homogeneous differential Equations Exact Differential Equations Making Exact by Integrating Factor First order Linear Differential Equation Chapter 2
Methods
Variable Separable Differential Equation Chapter 2 Separable Variable x is independent variable and y is dependent variable or Variable Separable Differential Equation Chapter 2 or are separable forms of the differential equation General solution can be solved by directly integrating both the sides + c Where c is constant of integration DO YOU REMEMBER INTEGRATION FORMULA? Chapter 2
Separation of Variables A differential equation of the type y’ = f(x)g(y) is separable. Definition Example Variable Separable Differential Equation Chapter 2 Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly. Example
Variable Separable Differential Equation Chapter 2 EXAMPLE: Variable Separable Differential Equation Chapter 2 Chapter 2
Variable Separable Differential Equation Chapter 2
Variable Separable Differential Equation Chapter 2
Note1: If we have Note.2. If we have Note.3. If we have Integrating by parts Note.2. If we have Integrating by parts Note.3. If we have Chapter 2
Variable Separable Differential Equation Chapter 2
Chapter 2
Chapter 2
Solution: Example: 12 Separable differential equation Combined constants of integration
Example (cont.): We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides.