Answers to Odd-Numbered Exercises
Chapter 0
Section 0.1
1.φ(x)=c 1 cos(λx)+c 2 sin(λx).- The equation has constant coefficientsk=0,p=0;u(t)=c 1 +c 2 t.
5.w(r)=c 1 rλ+c 2 r−λ. - Integrate, solve fordv/dx, and integrate again:
v(x)=c 1 +c 2 ln|h+kx|.
9.u(x)=c 1 +c 2 /x^2.
11.u(r)=c 1 +c 2 ln(r). - Characteristic polynomialm^4 −λ^4 =0; rootsm=±λ,±iλ. General so-
lutionu(x)=c 1 cos(λx)+c 2 sin(λx)+c 3 cosh(λx)+c 4 sinh(λx). - Characteristic polynomial(m^2 +λ^2 )^2 =0; rootsm=±iλ(double).
General solutionu(x)=(c 1 +c 2 x)cos(λx)+(c 3 +c 4 x)sin(λx).
17.v(t)=ln(t)andu 2 (t)=tbln(t).
19.u′′+λ^2 u=0;R(ρ)=(acos(λρ)+bsin(λρ))/ρ.
21.t^2 d^2 u/dt^2 =v′′−v′;tdu/dt=v′;v′′+(k− 1 )v′+pv=0 (constant
coefficients).
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