Manipulate[Plot[Sin[(Sqrt[λ])x], {x, 0, L}, PlotRange-> {{0,4.1}, {-1,1}}, PlotLabel -> "φ(x) = sin((Sqrt[λ])x), with Newton's Law of Cooling at x = L",AxesLabel ->{"x", "φ"},Prolog -> {
Line[{{L-.1,.99}, {L+.1, .81}}],
Line[{{L-.1,.77}, {L+.1, .63}}],
Line[{{L-.1,.55}, {L+.1, .45}}],
Line[{{L-.1,.33}, {L+.1, .27}}],
Line[{{L-.1,.11}, {L+.1, .09}}],
Line[{{L-.1,-.99}, {L+.1, -.81}}],
Line[{{L-.1,-.77}, {L+.1, -.63}}],
Line[{{L-.1,-.55}, {L+.1, -.45}}],
Line[{{L-.1,-.33}, {L+.1, -.27}}],
Line[{{L-.1,-.11}, {L+.1, -.09}}],
Line[{{L,1},{L,-1}}]}, AspectRatio -> Automatic ], {{λ, .67}, .4, 10, Appearance -> "Labeled"}, {{L, 3}, 1, 4, Appearance -> "Labeled"}]

When the figure first appears, it demonstrates that the first eigenvalue is (Subscript[λ, 1]) ≈ 0.67 when L = 3 in the eigenvalue problem
φ'' + λφ = 0
φ(0) = 0
φ'(L) + φ(L) = 0
You can move the λ slider and show that the next eigenvalue is (Subscript[λ, 2]) ≈ 3.0. 

Project Description:
It would be a good project to find the eigenvalues of this problem numerically.   You would want to start with the constants in Newton's law of cooling at x = L,
K φ'(L) + H φ(L) = 0,  equation (1.3.5)
and do scaling of x.  Show that there are not really 3 independent constants in the problem (K, H, and L) but the dimensionless constant (HL/K).  You will also find that the reasonable thing to compute numerically is λ(L^2).  Then, use a root finding function on a computer (NOT the demonstration above) to compute λ(L^2) as a function of HL/K and/or its reciprocal K/(HL).  You should consider the limit HL/K goes to 0 or infinity.  You might even consider the unphysical case of HL/K negative.  (See Section 5.8.)