Solutions

In 3.4., the solution will be real. The only source of complexity are the boundary conditions.

In[18]:=

(*3.4*)DSolve[{y''[t] + 3y '[t] - 2y[t] 1, y '[0] 1, y[0] 0}, y[t], t] Plot[y[t]/.%, {t, -1, 2}]

Out[18]=

{{y[t] 1/68 (-34 + 17 ^((-3/2 - 17^(1/2)/2) t) - 7 17^(1/2) ^((-3/2 ... /2)/2) t) + 17 ^((-3/2 + 17^(1/2)/2) t) + 7 17^(1/2) ^((-3/2 + 17^(1/2)/2) t))}}

[Graphics:../HTMLFiles/ex05_solutions_40.gif]

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⁃Graphics⁃

In[20]:=

(*3.5*)DSolve[{f''[x] + x f[x] 0, f '[0] 1, f[0] 2}, f[x], x ... #62513;0, f '[0] 1, f[0] 2}, f[x], {x, -3, 6}] Plot[f[x]/.%, {x, -3, 6}]

Out[20]=

{{f[x] 1/6 (3 (-1)^(2/3) 3^(1/3) AiryAi[(-1)^(1/3) x] Gamma[1/3] - (-1)^(2/3) 3^(5/6 ... [1/3] + 6 3^(2/3) AiryAi[(-1)^(1/3) x] Gamma[2/3] + 6 3^(1/6) AiryBi[(-1)^(1/3) x] Gamma[2/3])}}

Out[21]=

{{f[x] InterpolatingFunction[{{-3., 6.}}, <>][x]}}

[Graphics:../HTMLFiles/ex05_solutions_45.gif]

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⁃Graphics⁃

In[23]:=

(*3.6*)DSolve[x''[t] + (a - 2q Cos[2t]) x[t] 0, x[t], t]

Out[23]=

{{x[t] C[1] MathieuC[a, q, t] + C[2] MathieuS[a, q, t]}}

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