Homogeneous First Order Equations Bernd Schr oder logo1 Bernd - PowerPoint PPT Presentation

Overview An Example Double Check Homogeneous First Order Equations Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. 3. To solve a homogeneous first order equation, we translate the equation into a separable equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. 3. To solve a homogeneous first order equation, we translate the equation into a separable equation. 3.1 The substitution v = y x turns the homogeneous first order � y equation y ′ = f � into a separable equation for v , x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. 3. To solve a homogeneous first order equation, we translate the equation into a separable equation. 3.1 The substitution v = y x turns the homogeneous first order � y equation y ′ = f � into a separable equation for v , x 3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. 3. To solve a homogeneous first order equation, we translate the equation into a separable equation. 3.1 The substitution v = y x turns the homogeneous first order � y equation y ′ = f � into a separable equation for v , x 3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x , 3.3 After we solve the equation for v , we obtain y as xv . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check What are Homogeneous First Order Equations? 1. A homogeneous first order equation is of the form � y y ′ = f � . x 2. Recognizing homogeneous first order equations requires some pattern recognition. 3. To solve a homogeneous first order equation, we translate the equation into a separable equation. 3.1 The substitution v = y x turns the homogeneous first order � y equation y ′ = f � into a separable equation for v , x 3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x , 3.3 After we solve the equation for v , we obtain y as xv . That’s it. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y x , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y = vx ( !! ) x , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , y y y ′ = x − e x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , y y v ′ x + v = x − e x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , y v ′ x + v = v − e x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , v ′ x + v v − e v = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , v ′ x + v v − e v = v ′ x − e v = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Reduction to a separable equation. v : = y y ′ = v ′ x + v y = vx ( !! ) x , v ′ x + v v − e v = v ′ x − e v = − 1 v ′ xe v = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Solving the separable equation. dv − 1 xe v = dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Solving the separable equation. dv − 1 xe v = dx − 1 e − v dv = x dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Solving the separable equation. dv − 1 xe v = dx − 1 � � e − v dv = x dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Solving the separable equation. dv − 1 xe v = dx − 1 � � e − v dv = x dx − e − v = − ln | x | + c logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ = y y x − e x , y ( 1 ) = 1. Solving the separable equation. dv − 1 xe v = dx − 1 � � e − v dv = x dx − e − v = − ln | x | + c e − v = ln | x |− c logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous First Order Equations