Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations

\(y=-x+\dst{\frac{2}{e-1}\left(e^{x}-e^{(x-1)}\right)}\) 

\(y=x^{2}-\dst{\frac{x^{3}}{3}}+cx\) with \(c\) arbitrary

\(y=-x+2e^{x}+e^{-(x-1)}\) 

\(y= \dst{\frac{1}{4}+\frac{11}{4}\cos2x+\frac{9}{4}\sin2x}\)

\(y=(x^2+13-8x)e^{x}\) 

\(y=2e^{2x}+\dst{\frac{3(5e^{3x}-4e^{4x})}{e(15-16e)} +\frac{2e^{4}(4e^{3(x-1)}-3e^{4(x-1)})}{16e-15}}\)

\(\dst{\int_{a}^{b}tF(t)\,dt=0}\)  \(y=\dst{-x\int_{x}^{1}F(t)\,dt-\int_{0}^{x}tF(t)\,dt+c_{1}x}\) with \(c_{1}\) arbitrary

(a)  \(b-a\ne k\pi\)  (\(k=\) integer)

      \(y=\dst{\frac{\sin(x-a)}{\sin(b-a)}\int_{x}^{b}F(t)\sin(t-b)\,dt +\frac{\sin(x-b)}{\sin(b-a)}\int_{a}^{x}F(t)\sin(t-a)\,dt}\)

     (b)  \(\dst{\int_{a}^{b}F(t)\sin(t-a)\,dt=0}\)

        \(y=-\dst{\sin(x-a)\int_{x}^{b}F(t)\cos(t-a)\,dt -\cos(x-a)\int_{a}^{x}F(t)\sin(t-a)\,dt}\)

        \(+c_{1}\sin(x-a)\) with \(c_{1}\) arbitrary

(a)  \(b-a\ne(k+1/2)\pi\)  (\(k=\) integer)

        \(y=-\dst{ \frac{\sin(x-a)}{\cos(b-a)}\int_{x}^{b}F(t)\cos(t-b)\,dt- \frac{\cos(x-b)}{\cos(b-a)}\int_{x}^{b}F(t)\sin(t-a)\,dt}\)

     (b)  \(\dst{\int_{a}^{b}F(t)\sin(t-a)\,dt=0}\)

        \(y=-\dst{\sin(x-a)\int_{x}^{b}F(t)\cos(t-a)\,dt -\cos(x-a)\int_{a}^{x}F(t)\sin(t-a)\,dt}\)

        \(+c_{1}\sin(x-a)\) with \(c_{1}\) arbitrary

(a)  \(b-a\ne k\pi\) (\(k=\) integer)

        \(y=\dst{\frac{\cos(x-a)}{\sin(b-a)}\int_{x}^{b}F(t)\cos(t-b)\,dt+ \frac{\cos(x-b)}{\sin(b-a)}\int_{a}^{x}F(t)\cos(t-a)\,dt}\)

        (b) \(\dst{\int_{a}^{b}F(t)\cos(t-a)\,dt=0}\)

        \(y=\dst{\cos(x-a)\int_{x}^{b} F(t)\sin(t-a)\,dt +\sin(x-a)\int_{a}^{x} F(t)\cos(t-a)\,dt}\)

          \(+c_{1}\cos(x-a)\) with \(c_{1}\) arbitrary

\(y=\dst{\frac{\sinh(x-a)}{\sinh(b-a)}\int_{x}^{b}F(t)\sinh(t-b)\,dt +\frac{\sinh(x-b)}{\sinh(b-a)}\int_{a}^{x}F(t)\sinh(t-a)\,dt}\)

\(y=\dst{-\frac{\sinh(x-a)}{\cosh(b-a)}\int_{x}^{b}F(t)\cosh(t-b)\,dt -\frac{\cosh(x-b)}{\cosh(b-a)}\int_{a}^{x}F(t)\sinh(t-a)\,dt}\)

\(y=-\dst{\frac{\cosh(x-a)}{\sinh(b-a)}\int_{x}^{b}F(t)\cosh(t-b)\,dt -\frac{\cosh(x-b)}{\sinh(b-a)}\int_{a}^{x}F(t)\cosh(t-a)\,dt}\)

\(y=-\dst{\frac{1}{2}\left( e^{x}\int_{x}^{b}e^{-t}F(t)\,dt+e^{-x}\int_{a}^{x}e^{t}F(t)\,dt\right)}\)

If \(\omega\) isn’t a positive integer, then

       \(y=\dst{\frac{1}{\omega\sin\omega\pi} \left(\sin\omega x\int_{x}^{\pi}F(t)\sin\omega(t-\pi)\,dt +\sin\omega(x-\pi)\int_{0}^{x}F(t)\sin\omega t\,dt\right)}\).

     If \(\omega=n\) (positive integer), then \(\dst{\int_{0}^{\pi}F(t)\sin nt\,dt=0}\) is necessary for existence

     of a solution. In this case,

     \(y=-\dst{\frac{1}{n}\left(\sin nx\int_{x}^{\pi}F(t)\cos nt\,dt +\cos nx\int_{0}^{x} F(t)\sin nt\,dt\right) +c_{1}\sin nx}\)

     with \(c_{1}\) arbitrary.

If \(\omega\ne n+1/2\)  (\(n=\) integer), then

     If \(\omega= n+1/2\) (\(n=\) integer), then \(\dst{\int_{0}^{\pi}F(t)\sin(n+1/2)t\,dt=0}\) is necessary

     for existence of a solution. In this case,

     \(y=-\dst{\frac{\sin(n+1/2)x}{n+1/2}\int_{x}^{\pi}F(t)\cos(n+1/2)t\,dt}\)

           \(-\dst{\frac{\cos(n+1/2)x}{n+/2}\int_{0}^{x}F(t)\sin(n+1/2)t\,dt +c_{1}\sin(n+1/2)x}\)

     with \(c_{1}\) arbitrary,

If \(\omega\ne n+1/2\)  (\(n=\) integer), then

     If \(\omega= n+1/2\) (\(n=\) integer), then \(\dst{\int_{0}^{\pi}F(t)\cos(n+1/2)t\,dt=0}\) is necessary

     for existence of a solution. In this case,

     \(y=\dst{\frac{\cos(n+1/2)x}{n+1/2}\int_{x}^{\pi}F(t)\sin(n+1/2)t\,dt}\)

           \(+\dst{\frac{\sin(n+1/2)x}{n+/2}\int_{0}^{x}F(t)\cos(n+1/2)t\,dt +c_{1}\cos(n+1/2)x}\)

     with \(c_{1}\) arbitrary.

If \(\omega\) isn’t a positive integer, then

    \(y=\dst{\frac{1}{\omega\sin\omega\pi} \left(\cos\omega x\int_{x}^{\pi}F(t)\cos\omega(t-\pi)\,dt +\cos\omega(x-\pi)\int_{0}^{x}F(t)\cos\omega t\,dt\right)}\).

     If \(\omega=n\) (positive integer), then \(\dst{\int_{0}^{\pi}F(t)\cos nt\,dt=0}\) is necessary for existence

     of a solution. In this case,

     \(y=-\dst{\frac{1}{n}\left(\cos nx\int_{x}^{\pi}F(t)\sin nt\,dt +\sin nx\int_{0}^{x} F(t)\cos nt\,dt\right) +c_{1}\cos nx}\)

     with \(c_{1}\) arbitrary.

\(y_{1}=B_{1}(z_{2})z_{1}-B_{1}(z_{1})z_{2}\)

(a) \(G(x,t)= \begin{cases} \dst{\frac{(t-a)(x-b)}{b-a}} &a\le t\le x,\\[.11in] \dst{\frac{(x-a)(t-b)}{b-a)}}&x\le t\le b \end{cases}\)

     \(y=\dst{\frac{1}{b-a}\left((x-a)\int_{x}^{b}(t-b)F(t)\,dt+(x-b)\int_{a}^{x}(t-a)F(t)\,dt\right)}\)

   (b)  \(G(x,t)= \begin{cases} a-t & a\le t\le x\\ a-x &x\le t\le b \end{cases}\)   \(y=\dst{(a-x)\int_{x}^{b}F(t)\,dt+\int_{a}^{x}(a-t)F(t)\,dt}\)

   (c)  \(G(x,t)= \begin{cases} x-b & a\le t\le x\\ t-b &x\le t\le b \end{cases}\)   \(y=\dst{\int_{x}^{b}(t-b)F(t)\,dt+(x-b)\int_{a}^{x}F(t)\,dt}\)

   (d)  \(\dst{\int_{a}^{b}F(t)\,dt=0}\) is a necessary condition for existence of a solution. Then

        \(y=\dst{\int_{x}^{b}tF(t)\,dt+x\int_{a}^{x}F(t)\,dt+c_{1}}\) with \(c_{1}\) arbitrary.

\(G(x,t)= \begin{cases} \dst{-\frac{(2+t)(3-x)}{5}},& 0\le t\le x,\\[.11in] -\dst{\frac{(2+x)(3-t)}{5}}, &x\le t\le 1 \end{cases}\)   (a) \(y=\dst{\frac{x^{2}-x-2}{2}}\)   (b) \(y=\dst{\frac{5x^{2}-7x-14}{30}}\)

    (c)\(\, y=\dst{\frac{5x^{4}-9x-18}{60}}\)

\(G(x,t)= \begin{cases} \dst{\frac{\cos t\sin x}{t^{3/2}\sqrt{x}}}, &\dst{\frac{\pi}{2}}\le t\le x,\\[.11in] \dst{\frac{\cos x\sin t}{t^{3/2}\sqrt{x}}}, &x\le t\le \pi \end{cases}\)

      (a)  \(y=\dst{\frac{1+\cos x-\sin x}{\sqrt x}}\)   (b)  \(y=\dst{\frac{x+\pi \cos x-\pi/2\sin x}{\sqrt x}}\)

\(G(x,t)= \begin{cases} \dst{\frac{(t-1)x(x-2)}{t^{3}}},\; 1\le t\le x,\\[.11in] \dst{\frac{x(x-1)(t-2)}{t^{3}}},\; x\le t\le 2 \end{cases}\)

    (a) \(y=x(x-1)(x-2)\)  (b) \(y=x(x-1)(x-2)(x+3)\)

\(G(x,t)= \begin{cases} -\dst{\frac{1}{22}\left(3+\frac{1}{t^{2}}\right)\left(x+\frac{4}{x}\right)},&\; 1\le x\le t,\\[.11in] -\dst{\frac{1}{22}\left(3x+\frac{1}{x}\right)\left(1+\frac{4}{t^{2}}\right)},&\; x\le t\le 2 \end{cases}\)

    (a) \(y=\dst{\frac{x^{2}-11x+4}{11x}}\)  (b) \(y=\dst{\frac{11x^{3}-45x^{2}-4}{33x}}\)  (c) \(y=\dst{\frac{11x^{4}-139x^{2}-28}{88x}}\)

\(\alpha(\rho+\delta)-\beta\rho\ne0\)   \(G(x,t)= \begin{cases} \dst{\frac{(\beta-\alpha t)(\rho+\delta-\rho x)} {\alpha(\rho+\delta)-\beta\rho}}, \quad 0\le t\le x, \\[.11in] \dst{\frac{(\beta-\alpha x)(\rho+\delta-\rho t)} {\alpha(\rho+\delta)-\beta\rho}},\quad x\le t\le 1 \end{cases}\)

\(\alpha\delta-\beta\rho\ne0\)   \(G(x,t)= \begin{cases} \dst{\frac{(\beta\cos t-\alpha\sin t)(\delta \cos x-\rho\sin x)} {\alpha\delta-\beta\rho}},\; &0\le t\le x,\\[.11in] \dst{\frac{(\beta\cos x-\alpha\sin x)(\delta \cos t-\rho\sin t)} {\alpha\delta-\beta\rho}},\; &x\le t\le \pi \end{cases}\)

\(\alpha\rho+\beta\delta\ne0\)   \(G(x,t)= \begin{cases} \dst{\frac{(\beta\cos t-\alpha\sin t)(\rho\cos x+\delta\sin x)} {\alpha\rho+\beta\delta}}\; & x\le t\le \pi \\[.11in] \dst{\frac{(\beta\cos x-\alpha\sin x)(\rho\cos t+\delta\sin t)} {\alpha\rho+\beta\delta}}\; &0\le t\le x \end{cases}\)

\(\alpha\delta-\beta\rho\ne0\)   \(G(x,t)= \begin{cases} \dst{\frac{e^{x-t)}(\beta\cos t-(\alpha+\beta)\sin t) (\delta\cos x-(\rho+\delta)\sin x)}{\alpha\delta-\beta\rho}}\; &0\le t\le x, \\[.11in] \dst{\frac{e^{x-t}(\beta\cos x-(\alpha+\beta)\sin x) (\delta\cos t-(\rho+\delta)\sin t)}{\alpha\delta-\beta\rho}},\; &x\le t\le \pi \end{cases}\)

\(\beta \delta+(\alpha+\beta)(\rho+\delta)\ne0\)   \(G(x,t)= \begin{cases} \dst{\frac{e^{x-t}(\beta\cos t-(\alpha+\beta)\sin t) ((\rho+\delta)\cos x+\delta \sin x)} {\beta \delta+(\alpha+\beta)(\rho+\delta}},& 0\le t\le x, \\[.11in] \dst{\frac{e^{x-t}(\beta\cos x-(\alpha+\beta)\sin x) ((\rho+\delta)\cos t+\delta \sin t)} {\beta \delta+(\alpha+\beta)(\rho+\delta}}, &x\le t\le \pi/2 \end{cases}\)

\((\rho+\delta)(\alpha-\beta)e^{(b-a)}-(\rho-\delta)(\alpha+\beta)e^{(a-b)}\ne0\)

    \(G(x,t)= \begin{cases} \dst{\frac{((\alpha-\beta)e^{(t-a)}-(\alpha+\beta)e^{-(t-a)}) ((\rho-\delta)e^{(x-b)}-(\rho+\delta)e^{-(x-b)})} {2[(\rho+\delta)(\alpha-\beta)e^{(b-a)}-(\rho-\delta)(\alpha+\beta)e^{(a-b)}]}},\; &0\le t\le x,\\[.11in] \dst{\frac{((\alpha-\beta)e^{(x-a)}-(\alpha+\beta)e^{-(x-a)}) ((\rho-\delta)e^{(t-b)}-(\rho+\delta)e^{-(t-b)})} {2[(\rho+\delta)(\alpha-\beta)e^{(b-a)}-(\rho-\delta)(\alpha+\beta)e^{(a-b)}]}}\; &x\le t\le \pi \end{cases}\)

\((e^{bx}y')'+ce^{bx}y=0\) 

\(\dst{(xy')'+\left(x-\frac{\nu^{2}}{x}\right)y=0}\) 

\(\dst{(\sqrt{1-x^{2}}y')'+\frac{\alpha^{2}}{\sqrt{1-x^{2}}}y=0}\)

\((x^{b}y')'+cx^{b-2}y=0\) 

\((e^{-x^{2}}y')'+2\alpha e^{-x^{2}}y =0\) 

\((xe^{-x}y')'+\alpha e^{-x}y=0\)

\(((1-x^{2})y')'+\alpha(\alpha+1)y=0\) 

\(\lambda_{n}=n^{2}\pi^{2}\),  \(y_{n}=e^{-x}\sin n\pi x\)  (\(n=\) positive integer)

\(\lambda_{0}=-1\), \(y_{0}=1\) \(\lambda_{n}=n^{2}\pi^{2}\), \(y_{n}=e^{-x}(n\pi\cos n\pi x+\sin n\pi x)\) (\(n=\) positive

    integer)

(a) \(\lambda=0\) is an eigenvalue \(y_{0}=2-x\)  (b) none  (c) \(5.0476821\), \(14.9198790\),

    \(29.7249673\), \(49.4644528\) \(y=2\sqrt{\lambda}\cos\sqrt{\lambda}\,x-\sin\sqrt{\lambda} x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-0.5955245\) \(y=\cosh \sqrt{-\lambda}\,x\)  (c) \(8.8511386\),

    \(38.4741053\), \(87.8245457\), \(156.9126094\) \(y=\cos\sqrt\lambda x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) none  (c) \(0. 1470328\), \(1.4852833\), \(4.5761411\),

    \(9.6059439 \)  \(y=\sqrt\lambda\cos\sqrt\lambda x+\sin\sqrt\lambda x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-0.1945921\)

    \(y=2\sqrt{-\lambda}\,\cosh \sqrt{-\lambda}\,x\; -\sinh\sqrt{-\lambda}\,x\) (c) \(1.9323619\), \(5.9318981\), \(11.9317920\),

    \(19.9317507\) \(y=2\sqrt{\lambda}\,\cos \sqrt{\lambda}\,x-\sin\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-1.0664054\) \(y=\cosh\sqrt{-\lambda}\,x\)  (c) \(1.5113188\),

    \(8.8785880\), \(21.2104662\), \(38.4805610\) \(y=\cos\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-1.0239346\) 

    \(y=\sqrt{-\lambda}\,\cosh \sqrt{-\lambda}\,x\; -\sinh\sqrt{-\lambda}\,x\) (c) \(2.0565705\), \(9.3927144\), \(21.7169130\),

    \(38.9842177\)   \(y=\sqrt{\lambda}\,\cos \sqrt{\lambda}\,x-\sin\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-0.4357577\),

    \(y=2\sqrt{-\lambda}\,\cosh \sqrt{-\lambda}\,x\; -\sinh\sqrt{-\lambda}\,x\) (c) \(0.3171423\), \(3.7055350\), \(9.1970150\),

    \(16.8760401\)  \(y=2\sqrt{\lambda}\,\cos \sqrt{\lambda}\,x-\sin\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-2.1790546\), \(-9.0006633\)

    \(y=\sqrt{-\lambda}\,\cosh\sqrt{-\lambda}\,x-3\sinh\sqrt{-\lambda}\,x\)

    (c) \(5.8453181\), \(17.9260967\), \(35.1038567\), \(57.2659330\) \(y=\sqrt{\lambda}\,\cos\sqrt{\lambda}\,x-3\sin\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) is an eigenvalue  \(y_{0}=2-x\)  (b) \(-1.0273046 \)

    \(y=2\sqrt{-\lambda}\,\cosh\sqrt{-\lambda}\,x-\sinh\sqrt{-\lambda}\,x\) (c) \(8.8694608\), \(16.5459202\), \(26.4155505\),

    \(38.4784094\) \(y=2\sqrt{\lambda}\,\cos\sqrt{\lambda}\,x-\sin\sqrt{\lambda}\,x\)

(a) \(\lambda=0\) isn’t an eigenvalue  (b) \(-7.9394171\), \(-3.1542806\)

    \(y=2\sqrt{-\lambda}\,\cosh\sqrt{-\lambda}\,x-5\sinh\sqrt{-\lambda}\,x\) (c) \(29.3617465\), \(78.777456\), \(147.8866417\),

    \(236.7229622\) \(y=2\sqrt{\lambda}\,\cos\sqrt{\lambda}\,x-5\sin\sqrt{\lambda}\,x\)

\(\lambda=0\), \(y=xe^{-x}\) \(20.1907286\), \(118.8998692\), \(296.5544121\), \(553.1646458\)

    \(y=e^{-x}\sin\sqrt{\lambda}\,x\)

\(\lambda_{n}=n^{2}\pi^{2}\), \(y_{n}=x\sin n\pi(x-2)\)  (\(n=\) positive integer)

\(\lambda=0\), \(y=x(2-x)\) \(20.1907286\), \(118.8998692\), \(296.5544121\)

    \(553.1646458\), \(y=x\sin\sqrt{\lambda}\,(x-2)\)

\(3.3730893\), \(23.1923372\), \(62.6797232\), \(121.8999231\), \(200.8578309\)

    \(y=x\sin\sqrt{\lambda}\,(x-1)\)

(a) \(-L<\delta<0\)  (b) \(\delta=-L\)

\(\lambda_{0}=-1/\alpha^{2}\) \(y_{0}=e^{-x/\alpha}\)   \(\lambda_{n}=n^{2}\), \(y_{n}=n\alpha \cos nx-\sin nx\),  \(n=1\), \(2\), …

(a) \(y=x-\alpha\)   (b) \(y=\alpha k\cosh kx-\sin kx\) (c) \(y=\alpha k\cos kx-\sin kx\)

(b) \(\lambda=-\alpha^{2}/\beta^{2}\) \(y=e^{-\alpha x/\beta}\)