3.5we - 3.5 REDUCTION OF ORDER KIAM HEONG KWA Consider a...

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3.5: REDUCTION OF ORDER KIAM HEONG KWA Consider a general second-order linear homogeneouos equation (1) y 00 + p ( t ) y 0 + q ( t ) y = 0 with continuous coefficients on an open interval I . Provided with a nonzero solution y 1 ( t ), we want to calculate the general solution y ( t ) of (1). In the process, a first-order linear equation will be obtained. The first-order equation, once solved, provides us with the general solution y ( t ) of (1). This method is known as the method of reduction of order since the problem of solving a second-order equation is reduced to a problem of solving a first-order equation. There are two approaches for reduction of order . In one of them, we simply assumes that the general solution is given by y ( t ) = y 1 ( t ) v ( t ), where v ( t ) is a function to be determined. The problem is then reduced to seeking the function v ( t ). The second approach makes use of Abel’s theorem that we discussed in section 3.3 to construct a first-order linear equation for y ( t ).
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