Math 3 Section 6 PDF
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\"Variation of Parameters\" method for solving differential equations.
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## Variation of Parameters This method is used if the right hand side is not constant, homogeneous. In the previous method, it is suitable for all problems * ** y" + ay' + by = f(x) ** * Condition: Coefficient of y" = 1 * Steps of solving * Find y<sub>c</sub> * Y = C<sub>1...
## Variation of Parameters This method is used if the right hand side is not constant, homogeneous. In the previous method, it is suitable for all problems * ** y" + ay' + by = f(x) ** * Condition: Coefficient of y" = 1 * Steps of solving * Find y<sub>c</sub> * Y = C<sub>1</sub> y<sub>1</sub> + C<sub>2</sub> y<sub>2</sub> * y<sub>p</sub> = u<sub>1</sub>(x) y<sub>1</sub> + u<sub>2</sub>(x) y<sub>2</sub> * Assume y<sub>p</sub> * u<sub>1</sub>(x) = ∫ W<sub>1</sub> / W dx, u<sub>2</sub>(x) = ∫ W<sub>2</sub> / W dx * W = | y<sub>1</sub> y<sub>2</sub> | = | y<sub>1</sub>' y<sub>2</sub>' | * W<sub>1</sub> = | 0 y<sub>2</sub> | = | y<sub>1</sub>' y<sub>2</sub>' | * W<sub>2</sub> = | y<sub>1</sub> 0 | = | y<sub>1</sub>' f(x) | ## Solve **y" + y = secx** * ** Solution ** * Y = Y<sub>c</sub> + Y<sub>p</sub> * ** y<sub>c</sub> ** * y" + y = 0 * r<sup>2</sup> + 1 = 0 * r = ±i * Y<sub>c</sub> = C<sub>1</sub> cosx + C<sub>2</sub> sinx * ** y<sub>p</sub> ** * y<sub>p</sub> = u<sub>1</sub> cosx + u<sub>2</sub> sinx * W = | cosx sinx | = | -sinx cosx | = cos<sup>2</sup>x + sin<sup>2</sup>x = 1 * W<sub>1</sub> = | 0 sinx | = | secx cosx | = 0 - sinx * secx = - sinx / cosx = -tanx * W<sub>2</sub> = | cosx 0 | = | -sinx secx | = cosx * secx = 1 * u<sub>1</sub> = ∫ W<sub>1</sub> / W dx = ∫ - tanx / 1 dx = -∫ tanx dx = -∫ sinx / cosx dx = +ln(cosx) * u<sub>2</sub> = ∫ W<sub>2</sub> / W dx = ∫ 1 / 1 dx = x * :: y<sub>p</sub> = u<sub>1</sub> y<sub>1</sub> + u<sub>2</sub> y<sub>2</sub> * y<sub>p</sub> = ln(cosx) * cosx + x * sinx ## Solve **y" - 2y' + y = e<sup>x</sup> / 1+x<sup>2</sup>** * ** Solution ** * y = y<sub>c</sub> + y<sub>p</sub> * ** y<sub>c</sub> ** * y" - 2y' + y = 0 * r<sup>2</sup> - 2r + 1 = 0 <=> 1 * r = 1 * Y<sub>c</sub> = C<sub>1</sub>e<sup>x</sup> + C<sub>2</sub>xe<sup>x</sup> * ** y<sub>p</sub> ** * y<sub>p</sub> = u<sub>1</sub>e<sup>x</sup> + u<sub>2</sub>xe<sup>x</sup> * W = | e<sup>x</sup> xe<sup>x</sup> | = | e<sup>x</sup> xe<sup>x</sup> + e<sup>x</sup> | = xe<sup>2x</sup> + e<sup>2x</sup> - xe<sup>2x</sup> - e<sup>2x</sup> = 2x * W<sub>1</sub> = | 0 xe<sup>x</sup> | = | e<sup>x</sup> xe<sup>x</sup> + e<sup>x</sup> | = - xe<sup>2x</sup> / 1+x<sup>2</sup> * W<sub>2</sub> = | e<sup>x</sup> 0 | = | e<sup>x</sup> e<sup>x</sup> / 1+x<sup>2</sup> | = e<sup>2x</sup> / 1+x<sup>2</sup> * u<sub>1</sub> = ∫ W<sub>1</sub> / W dx = ∫ - xe<sup>2x</sup> / (1+x<sup>2</sup>) * 2x dx = -∫ x / (1+x<sup>2</sup>) dx = -1/2 ∫ 2x / (1+x<sup>2</sup>) dx = -1/2 ln(1+x<sup>2</sup>) * u<sub>2</sub> = ∫ W<sub>2</sub> / W dx = ∫ e<sup>2x</sup> / (1+x<sup>2</sup>) * 2x dx = ∫ 1 / (1+x<sup>2</sup>) dx = tan<sup>-1</sup>x * :: y<sub>p</sub> = u<sub>1</sub> y<sub>1</sub> + u<sub>2</sub> y<sub>2 </sub> * y<sub>p</sub> = -1/2 ln(1+x<sup>2</sup>) * e<sup>x</sup> + tan<sup>-1</sup>x * xe<sup>x</sup> ## Solve **y" - 2y' + y= e<sup>x</sup> / x** * ** Solution ** * y = y<sub>c</sub> + y<sub>p</sub> * ** y<sub>c</sub> ** * y" - 2y' + y = 0 * r<sup>2</sup> - 2r + 1 = 0 * (r-1)(r-1) = 0 * r = 1 * Y<sub>c</sub> = C<sub>1</sub>e<sup>x</sup> + C<sub>2</sub>xe<sup>x</sup> * ** y<sub>p</sub> ** * y<sub>p</sub> = u<sub>1</sub>(x) e<sup>x</sup> + u<sub>2</sub>(x) xe<sup>x</sup> * W = | e<sup>x</sup> xe<sup>x</sup> | = | e<sup>x</sup> xe<sup>x</sup> + e<sup>x</sup> | = xe<sup>2x</sup> + e<sup>2x</sup> - xe<sup>2x</sup> - e<sup>2x</sup> = e<sup>2x</sup> * W<sub>1</sub> = | 0 xe<sup>x</sup> | = | e<sup>x</sup> xe<sup>x</sup> + e<sup>x</sup> | = 0 - e<sup>2x</sup> = -e<sup>2x</sup> * W<sub>2</sub> = | e<sup>x</sup> 0 | = | e<sup>x</sup> e<sup>x</sup> / x | = e<sup>2x</sup> / x * 0= e<sup>2x</sup> / x * u<sub>1</sub> = ∫ W<sub>1</sub> / W dx = ∫ -e<sup>2x</sup> / e<sup>2x</sup> dx = ∫ -1 dx = -x * u<sub>2</sub> = ∫ W<sub>2</sub> / W dx = ∫ e<sup>2x</sup> / x / e<sup>2x</sup> dx =∫ 1 / x dx = ln x * :: y<sub>p</sub> = u<sub>1</sub> y<sub>1</sub> + u<sub>2</sub> y<sub>2</sub> * y<sub>p</sub> = -x * e<sup>x</sup> + ln x * xe<sup>x</sup> ## Cauchy Euler * **Form of equation** * ax<sup>2</sup>y" + bxy' + cy = f(x) * **Transform the equation into a constant coefficient equation by using**: * x = e<sup>t</sup> * t = ln x * xy' = ӱ * x<sup>2</sup>y" = ÿ - ÿ ## Solve **x<sup>2</sup> y" - 2xy' - 4y = 0** * **Solution** * ÿ - ÿ - 2ӱ - 4y = 0 * ÿ - 3ӱ- 4y = 0 * r<sup>2</sup> - 3r - 4 = 0 * (r-4)(r+1) = 0 * r = 4 , r= -1 * y = C<sub>1</sub>e<sup>4t</sup> + C<sub>2</sub>e<sup>-t</sup> * y = C<sub>1</sub>e<sup>4lnx</sup> + C<sub>2</sub>e<sup>-lnx</sup> = C<sub>1</sub>e<sup>lnx<sup>4</sup></sup> + C<sub>2</sub>e<sup>lnx<sup>-1</sup></sup> = C<sub>1</sub>x<sup>4</sup> + C<sub>2</sub>x<sup>-1</sup> ## Solve **4x<sup>2</sup>y" + y = 0** * **Solution** * 4(ÿ - ÿ) + y = 0 * 4ÿ - 4ӱ + y = 0 * 4r<sup>2</sup> - 4r + 1 = 0 * r<sub>1</sub> = r<sub>2</sub> = 1/2 * y = C<sub>1</sub>e<sup>lnx/2</sup> + C<sub>2</sub>t * e<sup>lnx/2</sup> = C<sub>1</sub>e<sup>lnx<sup>1/2</sup></sup> + C<sub>2</sub>lnx * e<sup>lnx<sup>1/2</sup></sup> = C<sub>1</sub>x<sup>1/2</sup> + C<sub>2</sub>lnx * x<sup>1/2</sup> ## Solve **x<sup>2</sup>y" - 6y = ln x** * ** y(1) = 1** * ** y'(1) = 3** * ** Solution ** * Cauchy Euler * ÿ - ÿ - 6y = t * ** y<sub>c</sub> ** * r<sup>2</sup> - r - 6 = 0 <=> r = 3, r = -2 * y = C<sub>1</sub>e<sup>3t</sup> + C<sub>2</sub>e<sup>-2t</sup> * ** y<sub>p</sub> ** * f(t) = t -> y<sub>p</sub> = At + B * y'<sub>p</sub> = A * y"<sub>p</sub> = 0 * Substitute in the equation that applies to y<sub>p</sub> * ÿ - ÿ - 6y = t * 0 - A - GA - 6B = t * -GA = 1 -> A= -1/6 * -A - 6B = 0 -> B = 1 / 36 * .. y = y<sub>c</sub> + y<sub>p</sub> * y = C<sub>1</sub>e<sup>3t</sup> + C<sub>2</sub>e<sup>-2t</sup> - 1/6 * t + 1/ 36 * y = C<sub>1</sub>e<sup>3lnx</sup> + C<sub>2</sub>e<sup>2lnx</sup> - 1/6 * ln x + 1 /36 * y = C<sub>1</sub>e<sup>lnx<sup>3</sup></sup> + C<sub>2</sub>e<sup>lnx<sup>-2</sup></sup> - 1/6 * ln x + 1 /36 * y = C<sub>1</sub>x<sup>3</sup> + C<sub>2</sub>x<sup>-2</sup> - 1/6 * ln x + 1 /36 * ** y(1) = 1 ** * 1 = C<sub>1</sub> + C<sub>2</sub> - 0 + 1 / 36 -> C<sub>1</sub> + C<sub>2</sub> = 35 / 36 * ** y'(1) = 3 ** * 3 = 3C<sub>1</sub> - 2C<sub>2</sub> - 1/6 -> 3C<sub>1</sub> - 2C<sub>2</sub> = 19/6 * **Solve system of equations** * C<sub>1</sub> = 46/45 * C<sub>2</sub> = -1/20 ## Solve **x<sup>2</sup>y" - xy' + y = x<sup>2</sup>** * ** Solution ** * Cauchy * ÿ - ÿ - ÿ + y = (e<sup>t</sup>)<sup>2</sup> * ÿ - 2ÿ + y = e<sup>2t</sup> * ** y<sub>c</sub> ** * r<sup>2</sup> - 2r + 1 = 0 -> r = 1 , r = 1 * Y<sub>c</sub> = C<sub>1</sub>e<sup>t</sup> + C<sub>2</sub>te<sup>t</sup> * ** y<sub>p</sub>** * f(t) = e<sup>2t</sup> -> y<sub>p</sub> = Ae<sup>2t</sup> * y'<sub>p</sub> = 2Ae<sup>2t</sup> * y"<sub>p</sub> = 4Ae<sup>2t</sup> * Substitute in the equation. * 4Ae<sup>2t</sup> - 4Ae<sup>2t</sup> + Ae<sup>2t</sup> = e<sup>2t</sup> * A = 1 -> y<sub>p</sub> = e<sup>2t </sup> * .. y = y<sub>c</sub> + y<sub>p</sub> * y = C<sub>1</sub>e<sup>t</sup> + C<sub>2</sub>te<sup>t</sup> + e<sup>2t</sup> * y = C<sub>1</sub>x + C<sub>2</sub>lnx * x + e<sup>2lnx</sup> = C<sub>1</sub>x + C<sub>2</sub>lnx * x + x<sup>2</sup> ## R-L-C Circuit * **Li' + iR + q/C = E(t)** * **Lq" + Rq' + q/C = E(t)** * **Given RLC circuit:** * L = 1H, R = 10 ohms, C = 1/9 F * E(t) = 9 * sin(t) * **Assume q(0) = I(0) = 0** * **Solution** * q" + 10q' + 1/9q = 9 * sin(t) * q" + 10q' + 9q = 9 * sin(t) * **r<sup>2</sup> + 10r + 9 = 0 <=> r = -1, r = -9 ** * **q<sub>c</sub> = C<sub>1</sub>e<sup>-t</sup> + C<sub>2</sub>e<sup>-9t</sup> ** * **f(t) = 9 * sin(t)** * q<sub>p</sub> = A * sin(t) + B * cos(t) * q'<sub>p</sub> = A * cos(t) - B * sin(t) * q"<sub>p</sub> = - A * sin(t) - B * cos(t) * **q"<sub>p</sub> + 10 q'<sub>p</sub> + 9q<sub>p</sub> = 9 * sin(t)** * -A * sin(t) - B * cos(t) + 10 * A * cos(t) - 10B * sin(t) + 9A * sin(t) + 9B * cos(t) = 9 * sin(t) * **sin(t):** -A - 10B + 9A = o -> 8A - 10B = 9 * **cos(t):** -B + 10A + 9B = 0 -> 10A + 8B = 0 * A = 13 / 41 * B = - 45 / 82 * **:: q(t) = C<sub>1</sub>e<sup>-t</sup> + C<sub>2</sub>e<sup>-9t</sup> + 13/41 * sin(t) - 45 / 82 * cos(t)** * **q(0) = 0 ** * 0 = C<sub>1</sub> + C<sub>2</sub> +0 - 45 /82 -> C<sub>1</sub> + C<sub>2</sub> = 45 / 82 * **I(0) = 0: q'(0) = 0** * q' = -C<sub>1</sub>e<sup>-t</sup> - 9 C<sub>2</sub>e<sup>-9t</sup> + 12 / 41 * cos(t) + 45 / 82 * sin(t) * 0 = -C<sub>1</sub> - 9C<sub>2</sub> + 12/41 + 0 -> C<sub>1</sub> + 9C<sub>2</sub> = 12/41 * **C<sub>1</sub> = 9 / 16** * **C<sub>2</sub> = -9 / 656**
