Separable Differential Equations Bernd Schr oder logo1 Bernd Schr - PowerPoint PPT Presentation

Definition An Example Double Check Separable Differential Equations Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check What are Separable Differential Equations? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check What are Separable Differential Equations? 1. A separable differential equation is of the form y ′ = f ( x ) g ( y ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check What are Separable Differential Equations? 1. A separable differential equation is of the form y ′ = f ( x ) g ( y ) . 2. That is, a differential equation is separable if the terms that are not equal to y ′ can be factored into a factor that only depends on x and another factor that only depends on y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check What are Separable Differential Equations? 1. A separable differential equation is of the form y ′ = f ( x ) g ( y ) . 2. That is, a differential equation is separable if the terms that are not equal to y ′ can be factored into a factor that only depends on x and another factor that only depends on y . 3. The solution method for separable differential equations looks like regular algebra with the added caveat that we use integrals to undo the differentials dx and dy from y ′ = dy dx . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx dy = x dx e y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx 1 2 x 2 + c − e − y = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx 1 2 x 2 + c − e − y = − c − 1 e − y 2 x 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx 1 2 x 2 + c − e − y = C − 1 e − y 2 x 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx 1 2 x 2 + c − e − y = C − 1 e − y 2 x 2 = � � C − 1 2 x 2 = − y ln logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . dy xe y = dx � dy � = x dx e y � � e − y dy = x dx 1 2 x 2 + c − e − y = C − 1 e − y 2 x 2 = � � C − 1 2 x 2 = − y ln � � C − 1 2 x 2 = y − ln logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . The general solution is a family of functions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . The general solution is a family of functions. � � C − 1 2 x 2 y = − ln logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . The general solution is a family of functions. � � C − 1 2 x 2 y = − ln logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . The general solution is a family of functions. � � C − 1 2 x 2 y = − ln logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . ◮ But how do we know that the above is right? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . ◮ But how do we know that the above is right? Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . ◮ But how do we know that the above is right? Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution. ◮ But how do we know that we did not make any mistakes? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . ◮ But how do we know that the above is right? Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution. ◮ But how do we know that we did not make any mistakes? We don’t. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check Solve the differential equation y ′ = xe y . ◮ The general solution of the differential equation y ′ = xe y is � � C − 1 2 x 2 y = − ln . ◮ But how do we know that the above is right? Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution. ◮ But how do we know that we did not make any mistakes? We don’t. ◮ So we should always check the result. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check � � C − 1 2 x 2 Does y = − ln Really Solve the Differential Equation y ′ = xe y ? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations

Definition An Example Double Check � � C − 1 2 x 2 Does y = − ln Really Solve the Differential Equation y ′ = xe y ? ? xe y y ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separable Differential Equations