Variation of Parameters - Nonhomogeneous 2nd Order Differential Equations | Math With Janine
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3 سال پیش
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In this video tutorial, I
In this video tutorial, I demonstrate how to solve nonhomogeneous 2nd order differential equations using the method of variation of parameters.
In a previous video, we learned about the method of undetermined coefficients for solving nonhomogeneous 2nd order differential equations. However, undetermined coefficients can only be used if f(x) has cyclic derivatives (e.g. sin(x), cos(x)) or if f(x) decimates with derivatives (e.g. polynomials). If f(x) does not satisfy either of these two criteria, we must use variation of parameters to solve our differential equation.
Steps
1. Put differential equation in standard form.
2. Find complementary solution.
3. Compute Wronskian.
4. Find particular solution. yp = u1y1+u2y2 where u1'=-y2*f/W and u2'=y1*f/W. Integrate u1' and u2' to obtain u1 and u2.
5. General solution: y = yc + yp
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In a previous video, we learned about the method of undetermined coefficients for solving nonhomogeneous 2nd order differential equations. However, undetermined coefficients can only be used if f(x) has cyclic derivatives (e.g. sin(x), cos(x)) or if f(x) decimates with derivatives (e.g. polynomials). If f(x) does not satisfy either of these two criteria, we must use variation of parameters to solve our differential equation.
Steps
1. Put differential equation in standard form.
2. Find complementary solution.
3. Compute Wronskian.
4. Find particular solution. yp = u1y1+u2y2 where u1'=-y2*f/W and u2'=y1*f/W. Integrate u1' and u2' to obtain u1 and u2.
5. General solution: y = yc + yp
► Follow me:
instagram | Instagram: mathwithjanine
tiktok | TikTok: mathwithjanine
twitter | Twitter: mathwithjanine
pinterest | Pinterest: mathwithjanine
3 سال پیش
در تاریخ 1400/01/06 منتشر شده
است.
2,103
بـار بازدید شده