Click on sideways triangles to open them. Execute commands as you go, or later ones won't make sense if you try to execute them. Tinker with this. See what happens if you change something.
<Text-field style="Heading 1" layout="Heading 1">Maple can carry out "undetermined coefficients" calculations.</Text-field>Here, we illustrate two approaches to this. The first simply calls up Maple's "dsolve" code, and the machine discovers that the problem is a match for the method of undetermined coefficients, and back comes the answer. The user never sees any of the internal logic. This can have a downside, as you will see at the bottom of this section. 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But also, we could use undetermined coefficients for ourselves, and see how things go, while leaving the tedious part to the computer. 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This is our linear operator. We check that it gives the right output. We use the notation that linear operators act on functions. So instead of asking about linop(x^3), we ask about linop (the function taking x to x^3). 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiJUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=This takes us back to what we'd write. Next, we try "linop", our differential operator, on the expression we expect to give our answer. We just don't know, yet, the values of a and b. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=And now we have an expression that ought to work out to e^(3x)sin(2x). The only way that will happen is if the coefficient combination 16a+18b on e^(3x)sin(2x) gives 1e^(3x)sin(2x), and the other combination gives 0. So, we ask Maple's ORDINARY "solve" operation to grind out the numbers:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GNjYmLUYjNjUtSSNtbkdGJDYkUSMxNkYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRkIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkovJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWS1GLDYlUSJhRidGL0YyLUZFNi1RKCZtaW51cztGJ0ZCRkhGS0ZNRk9GUUZTRlUvRlhRLDAuMjIyMjIyMmVtRicvRmVuRl1vLUY/NiRRIzE4RidGQkZELUYsNiVRImJGJ0YvRjItRkU2LVEiPUYnRkJGSEZLRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GZW5GaW8tRj82JFEiMUYnRkItRkU2LVEiLEYnRkJGSC9GTEYxRk1GT0ZRRlNGVUZXL0ZlblEsMC4zMzMzMzMzZW1GJ0Zfb0ZERmZuLUZFNi1RIitGJ0ZCRkhGS0ZNRk9GUUZTRlVGXG9GXm9GPkZERmJvRmVvLUY/NiRRIjBGJ0ZCRkIvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGXnAtRjY2Ji1GIzYlRmZuRl5wRmJvRkJGanBGXXFGQi1GRTYtUSI7RidGQkZIRmFwRk1GT0ZRRlNGVUZXRmpvThis agrees with what we got by thinking of Maple as an oracle, which gives answers only, with no shred of a reason for its answer. In the next line, another way to express the SELFSAME linear operator is given. That is, linop2 is mathematically identical to linop. Given any input, the two will return the same outputs as each other. 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Now let's see how this works for solving y''+3y'+2y=x^3. Our FORM is a polynomial of degree 3. So, we plug one in and see what comes out: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 we just need the coefficient on x^3 to work out to 1, and the others, to work out to zero. Maple might know about coefficients. After some hunting through the help, we come across the term "coeff", and if you execute the next line, you'll see how the command works. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEmY29lZmZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRic=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QyQtSSZjb2VmZkclKnByb3RlY3RlZEc2JC1JIl5HRiU2JEkieEc2IiIiIUkidkdGKyIiIg==Note that the "coeff" command doesn't give a sensible answer for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnZHNvbHZlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjAtRiw2JVElZGlmZkYnRi9GMi1GNjYkLUYjNigtRiw2JVEieUYnRi9GMi1GNjYkLUYjNiMtRiw2JVEieEYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGSy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZTLyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0ZILUZONi1RIiRGJ0ZLRlEvRlVGU0ZWRlhGWkZmbkZobkZqbi9GXm9GXG8tSSNtbkdGJDYkUSIyRidGS0ZLLUZONi1RIitGJ0ZLRlFGY29GVkZYRlpGZm5GaG4vRltvUSwwLjIyMjIyMjJlbUYnL0Zeb0ZdcC1GZm82JFEiM0YnRkstRk42LVEnJnNkb3Q7RidGS0ZRRmNvRlZGWEZaRmZuRmhuRmpuRmRvRjotRjY2JC1GIzYmRkFGREZNRkhGS0Zpb0Zlb0ZicEZBRkQtRk42LVEiPUYnRktGUUZjb0ZWRlhGWkZmbkZobi9GW29RLDAuMjc3Nzc3OGVtRicvRl5vRl1xLUklbXN1cEdGJDYlRkgtRiM2I0ZfcC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGSy1GTjYtUSI7RidGS0ZRRlRGVkZYRlpGZm5GaG5Gam5GXnE=One last example. Let's have our black box, dsolve, solve a similar problem but one unsuited to undetermined coefficients. We'll take g=1/x instead of something with the UDC Seal Of Approval. QyQtSSdkc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8sKC1JJWRpZmZHRiY2JC1JInlHRig2I0kieEdGKC1JIiRHRiY2JEYyIiIjIiIiLUYtNiRGL0YyIiIkRi9GNiokRjIhIiJGNw==What have we here? This is a fine example of the drawbacks of using a black box. In the next section, you will see the same problem worked, but with the gears out in the open, where you can see what happened and what the answer looks like. It should come as no great surprise to learn that the solution has a singularity at x=0. After all, so does the problem. JSFH
<Text-field style="Heading 1" layout="Heading 1">An example of "variation of parameters".</Text-field>Suppose we want to solve y''+3y'+2y=1/x. The method of undetermined coefficients does not apply. But for variation of parameters, all we need is to know y_1 and y_2, and to have the stomach to accept, as useful functions, functions given as integrals. The idea is we write y_p=v_1y_1+v_2y_2, leaving it to some algebra and calculus to work out suitable choices of NONCONSTANT functions v_1 and v_2. Since we are looking for one thing, y_p, and we have two functions, v_1 and v_2, we may reasonably expect that there are many suitable choices of the v's, and we can narrow the field without giving up hope that there will be a suitable choice in that narrower field. We're going to look under the streetlamp, where the light is better, because we expect that there are many keys, and we have a better chance of finding the keys that are under a light. Our light, here, is the simplifying assumption that v_1'y_1+v_2'y_2 is identically zero. This makes all our algebra and calculus easier, because when we compute the first derivative of y_p, we get only two terms, not four: v_1y_1'+v_2y_2'=y_p'. And when we compute y_p'', we get another lucky break: there are no terms involving v_1'' or v_2''. Consider our specific problem. Here, y_1 is the function taking x to e^(-x), and y_2, the function taking x to e^(-2x). For short, for working on paper, we'll just say y_1=e^(-x) and y2=e^(-2x). Our linear differential operator L=D^2+3D+2I takes an input function y and returns Ly=y''+3y'+2y. We want Ly_p=1/x. Number crunching time. This works out to the following system of equations: v_1'y_1+v_2'y_2=0; v_1'y_1'+v_2'y_2'=1/x, or plugging in what y_1 and y_2 are, it works out that we have two linear equations in the two unknowns w_1 and w_2, where w_1=v_1' and w_2=v_2': QyQ+SSt2cHJpbWVlcW5zRzYiNyQvLCYqJkkjdzFHRiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkSSJ4R0YlISIiRitGKyomSSN3MkdGJUYrLUYtNiMsJEYzISIjRitGKyIiIS8sJkYpRjRGNUY6KiRGM0Y0Ris=QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSt2cHJpbWVlcW5zR0YoNyRJI3cxR0YoSSN3MkdGKCIiIg==QyQ+SSN2MUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiYtSSRleHBHRi42I0kidEdGJSIiIkY2ISIiL0Y2O0Y3OSRGJUYlRiVGNw==QyQ+SSN2Mkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJi1JJGV4cEdGLzYjLCRJInRHRiUiIiMiIiJGOCEiIi9GODtGOjkkRjtGJUYlRiVGOg==QyQtSSN2MUc2IjYjSSJ4R0YlIiIiQyQ+SSN5cEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkOSQhIiIiIiItSSN2MUdGJTYjRjVGN0Y3KiYtRi82IywkRjUhIiNGNy1JI3YyR0YlRjpGN0Y3RiVGJUYlRjc=QyQ+SSZsb2Z5cEc2Ii1JJmxpbm9wR0YlNiNJI3lwR0YlIiIiQyQtSSZsb2Z5cEc2IjYjSSJ4R0YlIiIiQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIiQyQtSSVwbG90RzYiNiQtSSN5cEdGJTYjSSJ4R0YlL0YqOyMiIiIiIiQiIiNGLg==QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSN5cEdGKDYjSSJ4R0YoL0YtOyMiIiIiIikiIiRGMQ==QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSN5cEdGKDYjSSJ4R0YoL0YtOyIiIiIjP0YwEvidently, the fact of having an unevaluated integral is no great bar to further computing, to graphing, or to numerical evaluation. Maple has code that allows it to numerically evaluate integrals rather efficiently. QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JI3lwRzYiNiMiIiUiIiI=JSFH