Linear First Order Differential Equations Bernd Schr oder logo1 - PowerPoint PPT Presentation

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

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

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. � p ( x ) dx , 3.1 Compute the integrating factor µ ( x ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. � p ( x ) dx , 3.1 Compute the integrating factor µ ( x ) = e 3.2 Multiply the equation by the integrating factor, logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. � p ( x ) dx , 3.1 Compute the integrating factor µ ( x ) = e 3.2 Multiply the equation by the integrating factor, � p ( x ) dx y ′ + e � p ( x ) dx p ( x ) y note that the left side e � p ( x ) dx y, is the derivative of the product e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. � p ( x ) dx , 3.1 Compute the integrating factor µ ( x ) = e 3.2 Multiply the equation by the integrating factor, � p ( x ) dx y ′ + e � p ( x ) dx p ( x ) y note that the left side e � p ( x ) dx y, is the derivative of the product e 3.3 Integrate both sides, solve for y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check What are Linear First Order Differential Equations? 1. A linear first order differential equation is of the form y ′ + p ( x ) y = q ( x ) . 2. Recognizing linear first order differential equations requires some pattern recognition. 3. To solve a linear first order differential equation, we turn the left side into the derivative of a product. � p ( x ) dx , 3.1 Compute the integrating factor µ ( x ) = e 3.2 Multiply the equation by the integrating factor, � p ( x ) dx y ′ + e � p ( x ) dx p ( x ) y note that the left side e � p ( x ) dx y, is the derivative of the product e 3.3 Integrate both sides, solve for y . That’s it. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � p ( x ) dx = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) � p ( x ) dx = sin ( x ) dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) � p ( x ) dx = u : = sin ( x ) , sin ( x ) dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx du dx = cos ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx � cos ( x ) du du = dx = u cos ( x ) cos ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx � cos ( x ) du du = dx = u cos ( x ) cos ( x ) � 1 = u du logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx � cos ( x ) du du = dx = u cos ( x ) cos ( x ) � 1 = u du = ln | u | + c logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx � cos ( x ) du du = dx = u cos ( x ) cos ( x ) � 1 = u du = ln | u | logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

Overview An Example Double Check Solve the Initial Value Problem y ′ + cos ( x ) sin ( x ) y = 1, y ( 1 ) = 0. Integrating factor: � cos ( x ) du � p ( x ) dx = u : = sin ( x ) , dx = cos ( x ) , sin ( x ) dx � cos ( x ) du du = dx = u cos ( x ) cos ( x ) � 1 = u du = ln | u | � � = � sin ( x ) ln � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations