Eg. Solve y”-2yy’=0. (Sol.) Set u=y’, Eg. Solve . (Sol.) Let u=x2, Another method: (Bernoulli’s equation) Given a solution of then a second solution y1(x) y”+P(x)y’+Q(x)y=0, is obtained by the following method: y2(x)=v(x)y1(x) . Set v’=u Eg. Solve if is given. (Sol.) Let , Eg. is a solution of find the other solution. y(x)=x y”-xy’+y=0,
Eg. Solve y”-2yy’=0. (Sol.) Set u=y’, Eg. Solve . (Sol.) Let u=x2, Another method: (Bernoulli’s equation) Given a solution y1(x) of y”+P(x)y’+Q(x)y=0, then a second solution y2(x)=v(x)y1(x) is obtained by the following method: . Set v’=u Eg. Solve if is given. (Sol.) Let , Eg. y(x)=x is a solution of y”-xy’+y=0, find the other solution.
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