Differential Equations

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Presentation transcript:

Differential Equations Verification Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Verify that the given function satisfies the differential equation. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Verify that the given function satisfies the differential equation. First compute the derivative: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

 This is true, so the diff. eq. is satisfied Verify that the given function satisfies the differential equation. First compute the derivative: Now substitute into the differential equation:  This is true, so the diff. eq. is satisfied Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Substitute into diff. eq.: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

 It all cancels out, so the equation is satisfied Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Substitute into diff. eq.:  It all cancels out, so the equation is satisfied Now we need to satisfy the initial condition. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

 It all cancels out, so the equation is satisfied Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Substitute into diff. eq.:  It all cancels out, so the equation is satisfied Now we need to satisfy the initial condition. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

 It all cancels out, so the equation is satisfied Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Substitute into diff. eq.:  It all cancels out, so the equation is satisfied Now we need to satisfy the initial condition. Set this equal to -1 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

 It all cancels out, so the equation is satisfied Verify that the given function satisfies the differential equation, and find a value for C that satisfies the given initial condition. Find the derivative: Substitute into diff. eq.:  It all cancels out, so the equation is satisfied Now we need to satisfy the initial condition. Set this equal to -1 Our final solution is We will call this a particular solution because it matches with a particular initial condition. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB