Non-Homogeneous Second-Order Differential Equations James RohalWe are going to use Maple to solve some second-order differential equations. They take the form 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. A non-homogeneous second-order differential equations has LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSUmbmU7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRj1GPQ==. We are going to look at a special case where 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRiw2JVEiYkYnRi9GMkY1RlAtRiw2JVEiY0YnRi9GMkY5 are constants. The given by 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= is the complementary solution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= is the particular solution. The example we will be considering is 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. In an earlier worksheet we found that 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 by studying the characteristic polynomial. Below we show how to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= for different LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==.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If polynomial has degree n, then we guess that 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 where 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 are constants. Suppose 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. The degree is 2 so we guess 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. (It doesn't matter what we call the constants.) Now we use the method of undetermined coefficients. First, we find the derivatives of the particular solution.yp := A*x^2 + B*x + C;
yp_prime := diff(yp, x);
yp_double_prime := diff(yp, x$2);Now we substitute the expressions we found above in to the differential equation. ode := diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = x^2;
ode_with_yp := eval(ode, {diff(y(x), x$2) = yp_double_prime, diff(y(x), x) = yp_prime, y(x) = yp});We then collect common terms.collect(ode_with_yp, x);If the particular solution is in fact a solution to our ODE, then the coefficients on the LHS must agree with the coefficients on the RHS. This gives us a system of equations to solve.constants :=
solve([
(-6)*A + ( 0)*B + ( 0)*C = 1,
( 2)*A + (-6)*B + ( 0)*C = 0,
( 2)*A + ( 1)*B + (-6)*C = 0
],
{A,B,C});
Thus the particular solution is eval(yp, constants);The general solution is then 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.We can verify this using dsolve.dsolve(diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = x^2, y(x));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We guess that 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a constant. Suppose 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. Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk=, so we guess 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. Now we use the method of undetermined coefficients. First, we find the derivatives of the particular solution.yp := A*exp(3*x);
yp_prime := diff(yp, x);
yp_double_prime := diff(yp, x$2);Now we substitute the expressions we found above in to the differential equation. ode := diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = exp(3*x);
ode_with_yp := eval(ode, {diff(y(x), x$2) = yp_double_prime, diff(y(x), x) = yp_prime, y(x) = yp});If the particular solution is in fact a solution to our ODE, then the coefficients on the LHS must agree with the coefficients on the RHS. This says that 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. Thus the particular solution is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInBGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj1GPS1JI21vR0YkNi1RIj1GJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmluLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJ0Y9LUYjNiUtRmBvNiRRIjZGJ0Y9RjJGNS8lLmxpbmV0aGlja25lc3NHRmJvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxwLyUpYmV2ZWxsZWRHRlAtRks2LVEifkYnRj1GTkZRRlNGVUZXRllGZW4vRmhuUSYwLjBlbUYnL0Zbb0ZlcC1JJW1zdXBHRiQ2JS1GLzYlUSJlRidGMkY1LUYjNiYtRmBvNiRRIjNGJ0Y9RmFwRkdGPS8lMXN1cGVyc2NyaXB0c2hpZnRHRkFGPQ== and the general solution is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUklbXN1YkdGJDYlRistRiM2JC1GLDYlUSJjRidGL0YyRj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1LUZANi1RIitGJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGYG8tRlg2JUYrLUYjNiQtRiw2JVEicEYnRi9GMkY9RmluRjVGPy1GNjYkLUYjNigtRlg2JUZmbi1GIzYkLUkjbW5HRiQ2JFEiMUYnRj1GPUZpbi1JJW1zdXBHRiQ2JS1GLDYlUSJlRidGL0YyLUYjNiYtRmJwNiRRIjJGJ0Y9LUZANi1RIn5GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRJjAuMGVtRicvRlZGZHFGOkY9LyUxc3VwZXJzY3JpcHRzaGlmdEdGW29GXG8tRlg2JUZmbi1GIzYkRl1xRj1GaW4tRmZwNiVGaHAtRiM2Jy1GQDYtUSomdW1pbnVzMDtGJ0Y9RkNGRkZIRkpGTEZORlBGX29GYW8tRmJwNiRRIjNGJ0Y9RmBxRjpGPUZmcUY9Rj1GXG8tRjY2JC1GIzYmLUZANi1RIUYnRj1GQ0ZGRkhGSkZMRk5GUEZfb0Zhby1GIzYnRmpyLUkmbWZyYWNHRiQ2KEZhcC1GIzYkLUZicDYkUSI2RidGPUY9LyUubGluZXRoaWNrbmVzc0dGZHAvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGW3QvJSliZXZlbGxlZEdGRUZgcS1GZnA2JUZocC1GIzYmRmNyRmBxRjpGPUZmcUY9RmpyRj1GPUY9. We can verify this using dsolve.dsolve(diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = exp(3*x), y(x));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We guess that 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRiw2JVEiQkYnRi9GMkY5 are constants. Suppose 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. Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk= so we guess 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. Now we use the method of undetermined coefficients. First, we find the derivatives of the particular solution.yp := A*cos(3*x) + B*sin(3*x);
yp_prime := diff(yp, x);
yp_double_prime := diff(yp, x$2);Now we substitute the expressions we found above in to the differential equation. ode := diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = cos(3*x);
ode_with_yp := eval(ode, {diff(y(x), x$2) = yp_double_prime, diff(y(x), x) = yp_prime, y(x) = yp});We then collect common terms. We have to collect the terms with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiM0YnRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItRiw2JVEieEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMkYy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiM0YnRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItRiw2JVEieEYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMkYy separately.collected_ode := collect(ode_with_yp, cos(3*x));
collected_ode := collect(collected_ode, sin(3*x));If the particular solution is in fact a solution to our ODE, then the coefficients on the LHS must agree with the coefficients on the RHS. This gives us a system of equations to solve.constants :=
solve([
( -3)*A + (-15)*B = 0,
(-15)*A + ( 3)*B = 1
],
{A,B});
Thus the particular solution is eval(yp, constants);The general solution is then 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.We can verify this using dsolve.dsolve(diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = cos(3*x), y(x));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This is the most general case, as it is a product of cases above. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3RoRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn degree polynomial then our guess should be 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.
Suppose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUklbXN1cEdGJDYlLUYsNiVRImVGJ0YvRjItRiM2Jy1JI21uR0YkNiRRIjJGJ0Y9LUZANi1RIn5GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRJjAuMGVtRicvRlZGYW9GOkYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZANi1RJyZzZG90O0YnRj1GQ0ZGRkhGSkZMRk5GUEZgb0Zib0Y6RmZvLUYsNiVRJGNvc0YnL0YwRkVGPS1GNjYkLUYjNiYtRmpuNiRRIjNGJ0Y9Rl1vRjpGPUY9Rj0=. Here 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 so we guess 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. (I choose to use F because D is reserved by Maple). Now we use the method of undetermined coefficients. First, we find the derivatives of the particular solution.yp := exp(2*x)*(A*x+B)*cos(3*x) + exp(2*x)*(C*x+F)*sin(3*x);
yp_prime := diff(yp, x);
yp_double_prime := diff(yp, x$2);Now we substitute the expressions we found above in to the differential equation. ode := diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = exp(2*x)*x*cos(3*x);
ode_with_yp := eval(ode, {diff(y(x), x$2) = yp_double_prime, diff(y(x), x) = yp_prime, y(x) = yp});We then collect common terms. We have to collect the terms with 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, 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,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, 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 separately. This isn't the easiest thing in Maple so we do the best we can.collected_ode := collect(ode_with_yp, sin(3*x));
collected_ode := collect(collected_ode, cos(3*x));
collected_ode := collect(collected_ode, x);
collected_ode := collect(collected_ode, exp(2*x));If the particular solution is in fact a solution to our ODE, then the coefficients on the LHS must agree with the coefficients on the RHS. This gives us a system of equations to solve.constants :=
solve([
( -9)*A + ( 0)*B + (15)*C + ( 0)*F = 1,
(-15)*A + ( 0)*B + (-9)*C + ( 0)*F = 0,
( 5)*A + ( -9)*B + ( 6)*C + (15)*F = 0,
( -6)*A + (-15)*B + ( 5)*C + (-9)*F = 0
],
{A,B,C,F});
Thus the particular solution is eval(yp, constants);The general solution is then 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.We can verify this using dsolve. It turns out these solutions are the same.dsolve(diff(y(x), x$2) + diff(y(x), x) - 6*y(x) = exp(2*x)*x*cos(3*x), y(x));LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkNGLw==HomeworkSolve the differential equation 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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkNGLw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkNGLw==