Bernoulli Equations Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

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

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

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. 3. To solve a Bernoulli equation, we translate the equation into a linear equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. 3. To solve a Bernoulli equation, we translate the equation into a linear equation. 1 1 − n turns the Bernoulli equation 3.1 The substitution y = v y ′ + p ( x ) y = q ( x ) y n into a linear first order equation for v , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. 3. To solve a Bernoulli equation, we translate the equation into a linear equation. 1 1 − n turns the Bernoulli equation 3.1 The substitution y = v y ′ + p ( x ) y = q ( x ) y n into a linear first order equation for v , 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the 1 1 − n , substitution y = v logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. 3. To solve a Bernoulli equation, we translate the equation into a linear equation. 1 1 − n turns the Bernoulli equation 3.1 The substitution y = v y ′ + p ( x ) y = q ( x ) y n into a linear first order equation for v , 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the 1 1 − n , substitution y = v 3.3 After we solve the equation for v , we obtain y as the appropriate power of v . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check What are Bernoulli Equations? 1. A Bernoulli equation is of the form y ′ + p ( x ) y = q ( x ) y n , where n � = 0 , 1. 2. Recognizing Bernoulli equations requires some pattern recognition. 3. To solve a Bernoulli equation, we translate the equation into a linear equation. 1 1 − n turns the Bernoulli equation 3.1 The substitution y = v y ′ + p ( x ) y = q ( x ) y n into a linear first order equation for v , 3.2 We can even write down the abstract form of the resulting linear first order equation, but it is simpler to remember the 1 1 − n , substitution y = v 3.3 After we solve the equation for v , we obtain y as the appropriate power of v . That’s it. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. n = 5 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. 1 n = 5 , y = v 1 − 5 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. 1 − 5 = v − 1 1 n = 5 , y = v 4 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. 1 − 5 = v − 1 1 y ′ = n = 5 , y = v 4 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d 1 − 5 = v − 1 1 dxv − 1 n = 5 , y = v 4 , 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 y ′ + y x 2 y 5 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 � 5 y ′ + y x 2 � v − 1 = 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 � 5 y ′ + � v − 1 � x 2 � v − 1 = 4 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 − 1 � 5 4 v − 5 4 v ′ + � v − 1 � x 2 � v − 1 = 4 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 − 1 � 5 4 v − 5 4 v ′ + � v − 1 � x 2 � v − 1 = 4 4 − 1 4 v ′ + v − 1 4 v − 5 x 2 v − 5 = 4 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 − 1 � 5 4 v − 5 4 v ′ + � v − 1 � x 2 � v − 1 = 4 4 − 1 4 v ′ + v − 1 4 v − 5 x 2 v − 5 = 4 4 − 1 4 v ′ + v x 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + y = x 2 y 5 , y ( 0 ) = 1. Reduction to a linear equation. y ′ = d − 1 1 − 5 = v − 1 1 dxv − 1 4 v − 5 4 v ′ n = 5 , y = v = 4 , 4 − 1 � 5 4 v − 5 4 v ′ + � v − 1 � x 2 � v − 1 = 4 4 − 1 4 v ′ + v − 1 4 v − 5 x 2 v − 5 = 4 4 − 1 4 v ′ + v x 2 = v ′ − 4 v − 4 x 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bernoulli Equations