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## Reduction of Order

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**Reduction of Order**• Repeated Roots of the Characteristic Equation**So far…**• We’ve Learned How To Solve Second Order Linear Homogeneous ODEs with Constant Coefficients • Characteristic Equation • We’ve seen distinct real roots, complex conjugate roots**Characteristic Equation**Gives the Characteristic Equation Case 1: If roots are real and distinct, and General solution: Case 2: If roots are complex, and General solution:**Characteristic Equation**Gives the Characteristic Equation What if there’s only one root, and ?**Characteristic Equation**Gives the Characteristic Equation What if there’s only one root, Then we know one solution: How do we find another?**Reduction of Order**If we have a linear homogeneous second order equation and we know one solution How do we find another?**Reduction of Order**One solution In general, finding solutions is difficult**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE Can be solved with a first order linear ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE Make a clever “renaming”**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE Make a clever “renaming”**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE First Order Linear ODE**Reduction of Order**One solution Guess second solution has the form: Insert guess into ODE First Order Linear ODE Second solution is Use Wronskian to establish a Fundamental Set of Solutions**Important Example**Characteristic Equation: One Solution**Important Example**One Solution**Important Example**One Solution**Important Example**One Solution**Important Example**One Solution**Important Example**One Solution Second Solution**Important Example**One Solution: Second Solution: Wronskian: Wronskian: Form a Fundamental Set of Solutions General Solution:**This Holds in General**Gives the Characteristic Equation Only one root, and**This Holds in General**One solution: Determine Reduction of Order Equation**This Holds in General**One solution: Determine Reduction of Order Equation**This Holds in General**One solution: Determine Reduction of Order Equation**This Holds in General**One solution: Determine Reduction of Order Equation**This Holds in General**One solution: Determine Reduction of Order Equation Second Solution:**This Holds in General**If Characteristic Equation Has only one root, General Solution takes the form:**Summary**• We now know how to solve Second Order Linear Homogeneous with Constant Coefficients: • Characteristic Equation • Distinct Real Roots, Complex Conjugate Roots, One (Repeated) Root • If we have a Second Order Linear Homogeneous Equation and one solution, use Reduction of Order to find second solution.