is y1=t. Use the method of reduction of order to find a general solution of Initial Value Problem on the interval -1<t<1.
1
Expert's answer
2021-06-03T15:26:45-0400
To solve
(1−t2)(dt2d2y)+2t(dtdy)−2y=0
such that
y(0)=3, and y′(0)=−4
Let y(t)=tv(t),which gives dtdy(t)=tdtdv(t)+v(t)and dt2d2y(t)=tdt2d2v(t)+2dtdv(t)2t(tdtdv(t)+v(t))+(−t2+1)(tdt2d2v(t)+2dtdv(t))−2tv(t)=0Simplify:−tdt2d2v(t)(t2−1)+2dtdv(t)=0Let dtdV(t)=u(t),which gives dt2d2v(t)=dtdu(t):Solve for dtdu(t):dtdu(t)=t(t2−1)2u(t)Simplify:dtdu(t)=−−t3+t2u(t)
Divide both sides by u(t):u(t)dtdu(t)=−−t3+t2Integrate both sides with respect to t:∫u(t)dtdu(t)dt=∫−−t3+t2dtEvaluate the integrals:log(u(t))=−2(log(t)−21log(−t2+1))+c1,where c1is an arbitrary constant.Solve for u(t):u(t)=−t2ec1(t2−1)Simplify the arbitrary constants:u(t)=t2c1(t2−1)
Substitute back for dtdv(t)=u(t):dtdv(t)=t2c1(t2−1)Integrate both sides with respect to t :v(t)=∫t2c1(t2−1)dt=c1(t1+t)+c2, where c2 is an arbitrary constant.Substitute back for y(t)=tv(t), which gives v(t)=ty(t):ty(t)=c1(t1+t)+c2Solve for y(t):y(t)=c1+c1t2+c2tSolve for the unknown constants using the initial conditions:Compute dtdy(t):dtdy(t)=dtd(c1+c1t2+c2t)=2c1t+c2
Substitute y(0)=3 into y(t)=c1+t2c1+tc2:c1=3Substitute y′(0)=−4 into dtdy(t)=2tc1+c2:c2=−4Solve the system:c1=3c2=−4Substitute c1=3 and c2=−4 into y(t)=c1+t2c1+tc2:The required solution is:y(t)=3t2−4t+3
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