Use the method of reduction of order to find the general of the differential equation

midtlinjeg 2021-09-04 Answered

Use the method of reduction of order to find the general of the differential equation
t(t+3)y3(t+2)y+3y=0

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Expert Answer

Sally Cresswell
Answered 2021-09-05 Author has 91 answers

Solve td2y(t)dt2(t+3)3dy(t)dt(t+2)+3y(t)=0:
Let y(t)=(t+2)v(t), which gives dy(t)dt=dv(t)dt(t+2)+v(t)andd2y(t)dt2=d2v(t)dt2(t+2)+2dv(t)dt:3(t+2)(dv(t)dt(t+2)+v(t))+t(t+3)(d2v(t)dt2(t+2)+2dv(t)dt)+3(t+2)v(t)=0
Simplify:
td2v(t)dt2(t2+5t+6)+dv(t)dt(t26t12)=0
Let dv(t)dt=u(t), which gives d2v(t)dt2=du(t)dt:
Solve for du(t)dt:
du(t)dt=(t2+6t+12)u(t)t(t2+5t+6)

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