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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).

Using the differential operator D, the homogeneous equation y00 −y0 =0becomes D2 −D=0which has solutions D=1and D=0, corresponding to Dy= y(y= ex)andDy=0(y= constant). Thus, the general solution to the homogeneous equation is yh= c1 + c2ex.Wenowfind a particular solution to the original equation using undetermined coefficients.

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