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**Unformatted text preview: **nearly Dependent nearly I ndependent if A>0 and B>0 3: if A<0 4: if A>0 and B<0 =u1y1+u2y2 y1+u'2y2=0 y'1+u'2y'2=f(x) xWxdx+y2y1fxWxdx cosusinv sinusinv-cosA+B +cosA+B +sinA-B Bu2(A-B)-sinA+Bu2(A+B) Bu2(A-B)+sinA+Bu2(A+B) -Bu2(A-B)-cosA+Bu2(A+B) V x=d2ydV2dVdx+dydVddVdVdxdVdx VddV- =- e V1x d2ydV21x dydV1x1x 1 Linear Fi rst-Order Equations Integration factor: Exact Equations: + = ( ) dydx Pxy Q x + = M Ndydx 0 or + = Mdx Ndy 0 = x ePxdx Exact if: = M y N x = ( ) + yx 1 x y0 x0xtQtdt = ( - ) dPdt kP M P stable = ( - ) dPdt kP P M unstable Torricellis Law =- Aydydt a2gy 3.1 Second O rder Linear Diff erential Eq uation + ( ) + ( ) = ( ) y P x y Q x y F x Existence and uniqueness guaranteed if P(x) and Q(x) are continuous on interval I for initial = ya b0 and = y'a b1 then one solution for entire interval I Superposition: = + yc c1y1 c2y2 Characteristic Equation: + + = r2 Pr Q 0 Real Roots: = + yc C1er1x C2er2x Repeated Root: =( + ) yc C1 C2x erx Complex Roots: = ( + yc eax C1cosbx C2sinbx ) Wronskian- + = D a2 b2y 0 = r a bi = Wx fgf'g' 3.3 Homogenous Equations with Constant Coefficients = ++ + L andndxn...

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