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Solving Differential Equations: A Step-by-Step Guide, Study notes of Electrical Engineering
A detailed solution to a second-order differential equation, specifically (d^2+2d+1)y=xcosx. It begins by finding the complementary function (c.f) through the auxiliary equation and then proceeds to determine the particular integral (p.i). The solution involves techniques such as factoring, applying differential operators, and integrating trigonometric functions. This step-by-step approach provides a clear methodology for solving similar differential equations, making it a valuable resource for students studying calculus and differential equations. The document showcases the application of complex number theory and trigonometric identities in solving mathematical problems.
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Q) Solve the equation (D2+2D+1)y=xcosx where D=d/dx? Solve the equation (D2+2D+1)y=xcosx , where D=d/dx My Attempt: The given equation is (D2+2D+1)y=xcosx It's auxiliary equation is m^2 +2m+1= (m+1)^2 =0 m=−1,− Complementary Function (C.F)=(c1+c 2 x)e−x Now, the particular integral is P.I=xcosx/ D^2 +2D+ =x⋅cosx/D2+2D+1 – 2D+2 / (D2+2D+1)^2 x cosx =x⋅cosx/−1+2D+1 − 2(D+1)/ (D+1)4⋅cosx =x⋅cosx/2D − 2cosx/(D+1)^3 P.I= x⋅cosx/2D – 2 cosx/(D+1)^3 =xsinx/ 2 – 2 (D−1)^3 /(D2−1)^3 cosx =xsinx/ 2 – 2 /- 8 (D3−3D2+3D−1)cosx =xsinx/ 2 + ¼ [sinx+3cosx−3sinx−cosx] =xsinx2+12[cosx−sinx]