Soft versus Hard Springs. For Duffing's equation given in Problem 13, the behavior of the solutions changes as $r$ changes sign. When $r>0$, the restoring force $k y+r y^{3}$ becomes stronger than for the linear spring $(r=0)$. Such a spring is called hard. When $r<0$, the restoring force becomes weaker than the linear spring and the spring is called soft. Pendulums act like soft springs.

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(a) Redo Problem 13 with $r=-1$. Notice that for the initial conditions $y(0)=0, y^{\prime}(0)=1$, the soft and hard springs appear to respond in the same way for $t$ small.

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(b) Keeping $k=A=1$ and $\omega=10$, change the initial conditions to $y(0)=1$ and $y^{\prime}(0)=0$. Now redo Problem 13 with $r=\pm 1$.

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(c) Based on the results of part (b), is there a difference between the behavior of soft and hard springs for $t$ small? Describe.