1Chapter 2. 2 Example 3Chapter 2 4 EXAMPLE 5Chapter 2

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<ul><li> Slide 1 </li> <li> 1Chapter 2 </li> <li> Slide 2 </li> <li> 2 </li> <li> Slide 3 </li> <li> Example 3Chapter 2 </li> <li> Slide 4 </li> <li> 4 </li> <li> Slide 5 </li> <li> EXAMPLE 5Chapter 2 </li> <li> Slide 6 </li> <li> Solution 6Chapter 2 </li> <li> Slide 7 </li> <li> 7 </li> <li> Slide 8 </li> <li> 8 </li> <li> Slide 9 </li> <li> 9 Method for solving First Order Differential Equations Differential Equations Method for solving First Order Differential Equations Differential Equations </li> <li> Slide 10 </li> <li> Methods Variable Separable Reducible to variable separable Exact Differential Equation Integrating Factor </li> <li> Slide 11 </li> <li> Separable Variable x is independent variable and y is dependent variable or are separable forms of the differential equation or General solution can be solved by directly integrating both the sides + c Where c is constant of integration 11 Chapter 2 DO YOU REMEMBER INTEGRATION FORMULA </li> <li> Slide 12 </li> <li> Separation of Variables Definition A differential equation of the type y = f(x)g(y) is separable. Example Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly. </li> <li> Slide 13 </li> <li> EXAMPLE: 13Chapter 2 </li> <li> Slide 14 </li> <li> EXAMPLE: 14Chapter 2 </li> <li> Slide 15 </li> <li> To find the particular solution, we apply the given initial condition, when x =1, y = 3 is solution of initial value problem 15Chapter 2 </li> <li> Slide 16 </li> <li> 16Chapter 2 </li> <li> Slide 17 </li> <li> 17Chapter 2 </li> <li> Slide 18 </li> <li> 18 Note1: If we have Integrating by parts Note.2. If we have Integrating by parts Note.3. If we have </li> <li> Slide 19 </li> <li> Chapter 219 </li> <li> Slide 20 </li> <li> Chapter 220 </li> <li> Slide 21 </li> <li> Chapter 221 </li> <li> Slide 22 </li> <li> Method Homogeneous Equations Reducible to separable </li> <li> Slide 23 </li> <li> Chapter 223 Homogenous Differential Equations A differential equation Homogenous differential equation if every t, where t R </li> <li> Slide 24 </li> <li> Chapter 224 Example:1. Show that differential equation is homogenous differential equation. Solution: Differential equation is homogeneous Differential equation is homogeneous </li> <li> Slide 25 </li> <li> Chapter 225 METHOD for solving Homogenous differential equations Substitute OR </li> <li> Slide 26 </li> <li> Chapter 226 Using substitution the homogeneous differential equation is reduce to separable variable form. Example:2Solve the homogenous differential equation Solution: Rewriting in the form :. substitute and </li> <li> Slide 27 </li> <li> Chapter 227 is variable separable form is general solution. </li> <li> Slide 28 </li> <li> Chapter 228 Note.Selection of substitution Differential Equation depends on number of terms of coefficients 1.If, then take 2.If, then take 3.If, then take x = vy or y = ux </li> <li> Slide 29 </li> <li> Chapter 229 Example:.Solve the Differential Equation by using appropriate substitution Solution: Differential equation is homogeneous as degree of each term is same, hence we can use either y = ux or x = vy as substitution Substituting y and dy in the given equation (1 / 2) </li> <li> Slide 30 </li> <li> Chapter 230 is Separable form Integrating both the sides is general solution of the differential equation Separating variable u and x (2 / 2) </li> <li> Slide 31 </li> <li> Chapter 231 Example: Show that differential equation is homogeneous Solution: (1 / 2) </li> <li> Slide 32 </li> <li> Chapter 232 Let is general solution of the differential equation is Separable form Integrating both the sides (2 / 2) </li> <li> Slide 33 </li> <li> Chapter 233 Homogeneous Differential Equation Chapter 2 </li> <li> Slide 34 </li> <li> Chapter 234 (1 / 3) Homogeneous Differential Equation Chapter 2 </li> <li> Slide 35 </li> <li> Chapter 235 (2 / 3) Homogeneous Differential Equation Chapter 2 </li> <li> Slide 36 </li> <li> Chapter 236 (3 / 3) Homogeneous Differential Equation Chapter 2 </li> <li> Slide 37 </li> <li> Chapter 237 Homogeneous Differential Equation Chapter 2 </li> <li> Slide 38 </li> <li> Chapter 238 (1 / 2) Homogeneous Differential Equation Chapter 2 </li> <li> Slide 39 </li> <li> Chapter 239 is general solution of differential equation (2 / 2) Homogeneous Differential Equation Chapter 2 </li> <li> Slide 40 </li> <li> Chapter 240 Differential Equation Chapter 2 </li> <li> Slide 41 </li> <li> Chapter 241 Differential Equation Chapter 2 </li> <li> Slide 42 </li> <li> Chapter 242 is general solution of differential equation Differential Equation Chapter 2 </li> </ul>