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DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS

DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS1. Find the solution ofy0+2xy=x,withy(0) = is a linear equation. The integrating factor iseR2xdx=ex2. Multiplying through by this, we gety0ex2+2xex2y=xex2(ex2y)0=xex2ex2y=Rxe x2dx=12ex2+Cy=12+Ce in the initial condition givesC= 5/2,soy=12 52e= Find the general solution ofxy0=y (y2/x).A number of substitutions will work here. The simplest isy=ux,soy0=u+ gives a separable equation:x(u+u0x)=ux u2x2x=ux u2xdudxx2= u2xZ u 2du=Z1xdx1u=lnx+Cu=1lnx+Cy=xlnx+ Suppose that the frog populationP(t)of a small lake satisfies the DIFFERENTIAL equationdPdt=kP(200 P).(a) Find the equilibrium solutions. Sketch them and using the equation, sketch several solution curves,choosing some with initial points above and between the equilibrium equilibrium solutions areP=0(unstable) andP= 200(stable).

10. The reduced row echelon form for the matrix Abelow has been computed by Matlab: A= 2 −4 −12 −36 1−5 5 −10 −4 −1 rref(A)= 1 −203 00 14 00 00 Use this to find all solutions of 2x1 −4x2 −x3 =2 −3x1 +6x2 +x3 = −5 5x1 −10x2 −4x3 = −1 and express your answer in vector form.

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