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Question

Solve the differential equation (36D224D+13)y=2sin2xex+2

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Solution

The characteristic equation is 36p224p+13=0
p=b±b24ac2a
=24±2424(36)(13)72
(a=36,b=24,c=13)
24±129672=24±26i72=12(2±3i)72
=2±3i6=26+3i6=13+i2
Complementary functions is e13x[Acos12x+Bsinx2]....(1)
PI=2sin2xex+2=1cos2xex+2=3cos2xex[2.sin2x=1cos2x]
PI1=3.eax36D24D+13=3.eax00+13=313eax=313....(2)
PI2=ex36D224D+13=ex36(1)224(1)+13....(3)
=ex36+24+13=ex73
PI3=cos2x36D224D+13=cos2x36(4)24D+13
=cos2x14424D+13=cos2x24D131
=cos2x24D+131×24D13124D131=(24D+131)cos2x576D217161
=(24D+131)cos2x576(4)17161=(24D131)cos2x230417161
=(24D131)cos2x19465=24(2)(sin2x)131cos2x19465
=48sin2x131cos2x19465
=48sin2x+131cos2x19465
Solution is y=CF+PI1+PI2+PI3
y=ex3[Acosx2+Bsinx2]+313ex73+48sin2x+131cos2x19465

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