Transcription of DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS
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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).1(b) In the year 2000, its population was 100 and growing at the rate of 5 per year.
DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS 1. Find the solution of y0 +2xy= x,withy(0) = −2. This is a linear equation. The integrating factor is e R 2xdx= ex2. Multiplying through by this, we get y0ex2 +2xex2y = xex2 (ex2y)0 = xex2 ex2y = R xex2dx= 1 2 ex2 +C y = 1 2 +Ce−x2. Putting in the initial condition gives C= −5/2,soy= 1 2 ...
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