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50 Homework: Initial Value Problems

  1. Verify that the given solution to each differential equation is correct, and solve for the free parameter.
    1. Differential equation f(t)=f(t)+3, solution f(t)=Aet3, f(0)=4.
      f(t)=f(t)+3ddt(Aet3)=(Aet3)+3Aet=Aet.
      If f(0)=4, then A=7.

      ans
    2. Differential equation f(t)=2f(t)2, solution f(t)=Ae2t+1, f(0)=0.
      f(t)=2f(t)2ddt(Ae2t+1)=2(Ae2t+1)22Ae2t=2Ae2t+222Ae2t=2Ae2t
      If f(0)=0, then A=1.

      ans
    3. Differential equation f(x)=1f(x)+1, solution f(x)=A+2x+11, f(0)=4.
      f(x)=1f(x)+1ddx(A+2x+11)=1(A+2x+11)+1ddx(A+2x+1)1/2=1A+2x+112(A+2x+1)1/22=1A+2x+1(A+2x+1)1/2=1A+2x+11A+2x+1=1A+2x+1
      If f(0)=4, then A=24.

      ans
    4. Differential equation f(t)=(f(t))2+f(t), solution f(t)=AetAet1, f(0)=3.
      f(t)=(f(t))2+f(t)ddt(AetAet1)=(AetAet1)2+(AetAet1)(Aet1)AetAet(Aet)(Aet1)2=(Aet)2(Aet1)2AetAet1(Aet)2Aet(Aet)2(Aet1)2=(Aet)2(Aet1)2AetAet1(Aet1)(Aet1)Aet(Aet1)2=(Aet)2(Aet1)2(Aet)2Aet(Aet1)2Aet(Aet1)2=(Aet)2(Aet1)2+(Aet)2+Aet(Aet1)2Aet(Aet1)2=(Aet)2(Aet)2+Aet(Aet1)2Aet(Aet1)2=Aet(Aet1)2
      If f(0)=3, then A=1.5.

      ans

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