Solving a first order BVP with two boundary conditions with scipy's solve_bvp
<p>I am using scipy's BVP solver: </p>

<p><a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_bvp.html" rel="nofollow">http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_bvp.html</a></p>

<p>The problem I am running into is that you can only have as many boundary conditions as you have equations. I only have one equation but I have two boundary conditions. How can this be fixed? </p>

<p><strong>MWE</strong></p>

<pre><code>&gt;&gt;&gt; import numpy as np
&gt;&gt;&gt; from scipy.integrate import solve_bvp
&gt;&gt;&gt; 
&gt;&gt;&gt; x = np.linspace(0, 1, 100)
&gt;&gt;&gt; dydx = lambda x,y: y*np.sin(x)
&gt;&gt;&gt; 
&gt;&gt;&gt; result = solve_bvp(dydx, 
...     lambda ya,yb: np.array([ (ya[0]-1)**2 + (yb[0]-1)**2 ]),
...     x, [np.ones(len(x))], max_nodes=100000, tol=1e-9)
&gt;&gt;&gt; 
&gt;&gt;&gt; result
       message: 'The algorithm converged to the desired accuracy.'
         niter: 2
             p: None
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           sol: &lt;scipy.interpolate.interpolate.PPoly object at 0x2ad860930d58&gt;
        status: 0
       success: True
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         1.1226293 ,  1.12549819,  1.12838274,  1.13128299,  1.13419901,
         1.13713083,  1.14007851,  1.14304211,  1.14602166,  1.14901722,
         1.15202884,  1.15505658,  1.15810048,  1.1611606 ,  1.16423698,
         1.16732967,  1.17043874,  1.17356423,  1.17670619,  1.17986467,
         1.18303973,  1.18623141,  1.18943978,  1.19266488,  1.19590676,
         1.19916548,  1.20244108,  1.20573363,  1.20904318,  1.21236977,
         1.21571346,  1.2190743 ,  1.22245235,  1.22584765,  1.22926027,
         1.23269025,  1.23613766,  1.23960253,  1.24308492,  1.2465849 ,
         1.25010251,  1.2536378 ,  1.25719083]])
            yp: array([[ 0.        ,  0.00267302,  0.0053461 ,  0.0080193 ,  0.01069269,
         0.01336631,  0.01604024,  0.01871453,  0.02138925,  0.02406445,
         0.0267402 ,  0.02941655,  0.03209358,  0.03477132,  0.03744986,
         0.04012924,  0.04280954,  0.0454908 ,  0.04817309,  0.05085648,
         0.05354102,  0.05622677,  0.05891379,  0.06160215,  0.0642919 ,
         0.06698311,  0.06967583,  0.07237013,  0.07506607,  0.0777637 ,
         0.0804631 ,  0.08316431,  0.08586741,  0.08857244,  0.09127948,
         0.09398858,  0.0966998 ,  0.09941321,  0.10212887,  0.10484683,
         0.10756715,  0.11028991,  0.11301515,  0.11574295,  0.11847335,
         0.12120642,  0.12394223,  0.12668083,  0.12942228,  0.13216665,
         0.134914  ,  0.13766438,  0.14041786,  0.1431745 ,  0.14593436,
         0.1486975 ,  0.15146398,  0.15423387,  0.15700722,  0.1597841 ,
         0.16256456,  0.16534867,  0.16813649,  0.17092808,  0.1737235 ,
         0.17652281,  0.17932607,  0.18213335,  0.18494471,  0.1877602 ,
         0.1905799 ,  0.19340385,  0.19623212,  0.19906478,  0.20190187,
         0.20474348,  0.20758965,  0.21044044,  0.21329593,  0.21615617,
         0.21902122,  0.22189114,  0.22476599,  0.22764585,  0.23053076,
         0.23342079,  0.236316  ,  0.23921645,  0.2421222 ,  0.24503332,
         0.24794987,  0.2508719 ,  0.25379948,  0.25673268,  0.25967155,
         0.26261615,  0.26556655,  0.2685228 ,  0.27148497,  0.27445313,
         0.27742732,  0.28040762,  0.28339409,  0.28638678,  0.28938576,
         0.29239109,  0.29540283,  0.29842105,  0.3014458 ,  0.30447715,
         0.30751515,  0.31055988,  0.31361139,  0.31666974,  0.31973499,
         0.32280722,  0.32588647,  0.32897281,  0.3320663 ,  0.33516701,
         0.33827498,  0.3413903 ,  0.34451301,  0.34764319,  0.35078088,
         0.35392616,  0.35707908,  0.3602397 ,  0.3634081 ,  0.36658432,
         0.36976843,  0.37296049,  0.37616057,  0.37936872,  0.382585  ,
         0.38580948,  0.38904223,  0.39228329,  0.39553273,  0.39879061,
         0.402057  ,  0.40533195,  0.40861553,  0.4119078 ,  0.41520881,
         0.41851863,  0.42183733,  0.42516495,  0.42850157,  0.43184723,
         0.43520202,  0.43856597,  0.44193917,  0.44532166,  0.4487135 ,
         0.45211476,  0.45552551,  0.45894578,  0.46237566,  0.4658152 ,
         0.46926446,  0.47272349,  0.47619237,  0.47967114,  0.48315988,
         0.48665863,  0.49016747,  0.49368644,  0.49721562,  0.50075505,
         0.5043048 ,  0.50786493,  0.5114355 ,  0.51501656,  0.51860818,
         0.52221041,  0.52582331,  0.52944695,  0.53308138,  0.53672666,
         0.54038285,  0.54405001,  0.54772819,  0.55141745,  0.55511786,
         0.55882946,  0.56255232,  0.5662865 ,  0.57003205,  0.57378903,
         0.5775575 ,  0.58133751,  0.58512912,  0.58893239,  0.59274738,
         0.59657414,  0.60041272,  0.60426319,  0.6081256 ,  0.61200001,
         0.61588646,  0.61978503,  0.62369576,  0.6276187 ,  0.63155392,
         0.63550147,  0.6394614 ,  0.64343376,  0.64741862,  0.65141602,
         0.65542602,  0.65944867,  0.66348403,  0.66753215,  0.67159308,
         0.67566687,  0.67975358,  0.68385327,  0.68796597,  0.69209174,
         0.69623064,  0.70038272,  0.70454802,  0.7087266 ,  0.7129185 ,
         0.71712379,  0.7213425 ,  0.72557469,  0.72982041,  0.7340797 ,
         0.73835262,  0.74263921,  0.74693953,  0.75125361,  0.75558151,
         0.75992327,  0.76427895,  0.76864858,  0.77303222,  0.7774299 ,
         0.78184168,  0.7862676 ,  0.79070771,  0.79516204,  0.79963065,
         0.80411358,  0.80861086,  0.81312256,  0.81764869,  0.82218932,
         0.82674447,  0.8313142 ,  0.83589854,  0.84049753,  0.84511122,
         0.84973964,  0.85438283,  0.85904083,  0.86371368,  0.86840142,
         0.87310408,  0.8778217 ,  0.88255432,  0.88730198,  0.89206471,
         0.89684254,  0.90163551,  0.90644365,  0.911267  ,  0.91610559,
         0.92095945,  0.92582862,  0.93071312,  0.93561298,  0.94052825,
         0.94545894,  0.95040508,  0.95536671,  0.96034386,  0.96533654,
         0.97034479,  0.97536863,  0.98040809,  0.98546319,  0.99053396,
         0.99562042,  1.0007226 ,  1.00584051,  1.01097418,  1.01612363,
         1.02128888,  1.02646995,  1.03166686,  1.03687962,  1.04210827,
         1.0473528 ,  1.05261324,  1.0578896 ]])
</code></pre>

<p>As you can see, y is very far from the boundary conditions of <code>y(x=0) = y(x=1) = 1</code>.</p>

<p>If you specify two boundary conditions y(0)=1 and y(1)=1 for a first order ODE, then in general the problem is overdetermined and <em>there is no solution</em>. If you specify just the initial condition y(0)=y0, you have a first order initial value problem. In fact, in this case, you can derive the exact solution: y(x) = y0*exp(-cos(x)). </p>

python numpy scipy numerical-methods differential-equations