ODE Cheat Sheet Nonhomogeneous Problems Series Solutions

ODE Cheat SheetFirst order EquationsSeparabley (x) =f(x)g(y) dyg(y)= f(x)dx+CLinear First Ordery (x) +p(x)y(x) =f(x) (x) = exp xp( )d Integrating factor.( y) =f Exact :y(x) =1 (x)( f( ) ( )d +C)Exact0 =M(x,y)dx+N(x,y)dySolution:u(x,y) = const wheredu= u xdx+ u ydy u x=M(x,y), u y=N(x,y)Condition:My=NxNon-Exact Form (x,y) (M(x,y)dx+N(x,y)dy) =du(x,y)My=NxN x M y= ( M y N x).Special casesIfMy NxM=h(y),then (y) = exp h(y)dyIfMy NxN= h(x),then (y) = exp h(x)dxSecond order EquationsLineara(x)y (x) +b(x)y (x) +c(x)y(x) =f(x)y(x) =yh(x) +yp(x)yh(x) =c1y1(x) +c2y2(x)Constant Coefficientsay (x) +by (x) +cy(x) =f(x)y(x) =erx ar2+br+c= 0 CasesDistinct, real roots:r=r1,2, yh(x) =c1er1x+c2er2xOne real root:yh(x) = (c1+c2x)erxComplex roots:r= i , yh(x) = (c1cos x+c2sin x)e xCauchy-Euler Equationsax2y (x) +bxy (x) +cy(x) =f(x)y(x) =xr ar(r 1) +br+c= 0 CasesDistinct, real roots:r=r1,2, yh(x) =c1xr1+c2xr2 One real root:yh(x) = (c1+c2ln|x|)xrComplex roots:r= i ,yh(x) = (c1cos( ln|x|) +c2sin( ln|x|))x Nonhomogeneous ProblemsMethod of Undetermined Coefficientsf(x)yp(x)anxn+ +a1x+a0 Anxn+ +A1x+A0aebxAebxacos x+bsin xAcos x+Bsin xModified Method of Undetermined Coefficients.

Reduction of Order Homogeneous Case Given y 1(x) satis es L[y] = 0; nd second linearly independent solution as v(x) = v(x)y 1(x):z= v0satis es a separable ODE. Nonhomogeneous Case Given y 1(x) satis es L[y] = 0; nd solution of L[y] = fas v(x) = v(x)y 1(x):z= v0satis es a rst order linear ODE. Method of Variation of Parameters yp(x) = c 1(x)y 1 ...

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  Second, Order, Nonhomogeneous

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