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Differential EquationIndefinite IntegralsClass WorkFind ydydx=4x2+2, y2=5dydx=sinx, y0=6dydx=2x3-4x, y3=2dydx=1x+6x, y1=0dydx=x4+3x2+4x+5,y4=-1d2ydx2=6x+1, y'2=3, y2=1d2ydx2=1x2, y'13=4, y1=3d2ydx2=2, y2=6,y1=2Homework dydx=3x3+x, y1=-5dydx=cos2x, y0=4dydx=4x2-5x, y7=-2dydx=2x+8x, y1=4dydx=5x4-2x2+8x-9,y4=-1d2ydx2=3x+2, y'2=3, y2=1d2ydx2=-12x2, y'14=5, y1=7d2ydx2=6x, y2=3,y1=5Slope FieldsClass WorkSketch fields for the following differential equations using the lattice points on [-3,3] by [-3,3].y'=x, (1,1)y'=y, (2,1)y'=2-x, (2,3)y'=x-2, (2,3)y'=x-y, (0,1)y'=xy , (2,1)Homework Sketch fields for the following differential equations using the lattice points on [-3,3] by [-3,3]. Then sketch the solution that passes through the given point.y'=2x, (1,0)y'=2y, (0,1)y'=1y , (2,1)y'=1x , (2,1)y'=y-x, (0,0)y'=yx, (2,1)U-SubstitutionClass WorkEvaluate the integral.2x+52dx6x3x2-83dxcos4xdxxsinx2dxsin2xcos2x2dxsec3xtan3xdx1-4xdx43x+6dx3xx2-8dxxx+4dxHomework Evaluate the integral.4x-53dx4x5x2-24dx4cos3xdxxsin5x2dxcos2xsin2x2dxsec3x2dx5+6xdx-75x-8dx4x3x2+9dxxx+4dxU-Substitution & Definite IntegralsClass WorkEvaluate the integral.052x+32dx-142xx2-33dx473x-4dx0π4cos2xsin2x3dx0135x-6dx0ln5exex-22dx-102x1+x2dx274xx+3dxHomework Evaluate the integral.-235x-22dx-353x4-x23dx235x-6dx0π2xcosx2dx0136-5xdx-ln30e-xe-x+4dx-21x8-x2dx015x3-x2dxDifferential Equations (Separable First Order)Class WorkFind the general solution. If conditions are given find the particular solution.dydx=5x3ydydx=4x23y2drdt=2rtdydx=4xydydx=5x3y, y1=-4dydx=4x23y2, y=3 when x=6d2rdt2=2rt, r'=9 when t=4and r=2, r=3 when t=1d2ydx2=6xy2, y'=2 when x=3and y=-14;y=1 when x=1 Homework Find the general solution. If conditions are given find the particular solution.y'=4x5ydydx=6x28y3drdt=4r2tf'(x)=3xydydx=6xy, y2=4dydx=x3y3, y=-1 when x=2d2rdt2=7rt2, r'=4 when t=-12and r=3;r=6 when t=1dydx=2xey, y'=-5 when x=6and y=0;y=0 when x=0 Integration by PartsClass WorkFind the integral.xsinx dx4x e2xdxx3lnx dxx3cos2xdx 5x6e-xdxe2xcosxdxe-3xsin3xdx12x2ln2xdxHomework Find the integral.3xcosx dx2x e-xdx4x3ln3x dxx4sin3xdx 1120x5e2xdxexcos4xdxe4xsinxdx24x3lnxdxPopulation GrowthClass WorkWrite the specific functions by solving the initial value problems.dydx=3y and y1=2dydx=6xy and y1=-4A colony of bees grows exponentially. At the end of 2 hours there were 300. At the end of 5 hours there were 350. How many bees were there initially?An isotope decays exponentially. If there were 600g initially and in 10 hours there were 500g, what is the isotope’s half-life?A bacteria has a growth rate of dydx=10y, and there were 1000 initially, how long until there are 10,000 bacteria?100 Brook trout are released into a protected stream. If the population is measured to grow at the rate dPdx=P(700-P) where t is weeks, what is limx→∞Pt= ?Homework Write the specific functions by solving the initial value problems.dydx=5y and y0=-3dydx=4xy and y2=6A colony of bees grows exponentially. At the end of 3 hours there were 500. At the end of 5 hours there were 550. How many bees were there initially?An isotope decays exponentially. If there were 600g initially and in 50 hours there were 400g, what is the isotope’s half-life?A bacteria has a growth rate of dydx=5y, and there were 1000 initially, how long until there are 5,000 bacteria?200 Brook trout are released into a protected stream. If the population is measured to grow at the rate dPdt=P(600-P) where t is weeks, what is limx→∞Pt=?Multiple ChoiceCalculator Not Permitted3-6tdt=233-6t32+C-43-6t32+C-233-6t32+C-293-6t32+C1-93-6t32+C04x5-xdx=105-2314+5555-2837.55-91320-163403x2exdxx2ex+2xex+Cx2ex-2xex+2ex+Cx2ex+2xex+2ex+Cx2ex-2xex-2ex+Cx2ex+2xex+2ex+CIf dydx=yx2 and y1=-1 thenee-1x-ee-1xee1x-ee1xe2xIf d2ydx2=6x, with y'2=1 and y1=2, then y= y=x3-9x+10y=x3-9x-8y=x3-11x-10y=x3-11x+12y=x3-9x+12Calculator Permittedh(x)=f(g(x)) then h(x)dx=F(g(x))+CF(g(x))/g’(x)+Cfuddug-1(uduf(u)g'(u)-1duF(u)G-1(u)Suppose at the age of 20 you invest at a rate of 5% compounded continuously. If you want the account to have $1,000,000 when you’re 60, what should be the investment? $49,787$68,918$135,335$151,719$180,023e2xcos3x dx=12cos3x e2x+34sin3x e2x+C12cos3x e2x-34sin3x e2x+C14cos3x e2x+38sin3x e2x+C112cos3x e2x+116sin3x e2x+C213cos3x e2x+313sin3x e2x+CWhen yπ4=2, sec2x dx=tan x +1tan x +3sec x +1sec xsec x tan xThe given slope field is the for the differential equation2335530-76200dydx=1xdydx=2xdydx=1ydydx=1ydydx=xy419989036194Extended ResponseCalculator Not PermittedGiven the differential equation of dydx=2yxdraw the slope fielddraw the solution that passes through (1,0)Integrate to find the equation of the curve through (1,0)15x3x-2dxCalculator PermittedA national parks moose population is growing at a rate directly proportional to (800 – P) Find P(t) in terms of t (time), k, and C.Find P(10) if P(0)=200 and P(2)=300.Find limt→∞P(t)Find y if d2ydx2=x2y and y’(1)=4 and y(2)=93554730158750Answer KeyY= 4/3x2+2x-29/3Y=-cosx+7Y=1/2x4-2x2+20.5Y=lnx+3x2-3Y=1/5x5+x3+2x2+5x-341.8Y=x3+1/2x2-11x+13Y=-ln|x|+7x-4Y=x2-x+1Y=3/4x4+1/2x2-6.25Y=1/2sin2x+3.5Y=4/3x3-5/2x2 – 2021/6Y=2ln|x|+4x2Y=x5-2/3x2+4x2-9x-1010.3331/2x3+x2-7x+7Y=1/2ln|x|+3x+4Y=x3-9x+13354330247015328930306070476250-635476250127047625018161047625004203701403344203701079504102093873539497097155445770-1066801/6(2x+5)2+c?(3x2-8)4+c1/4sin4+c-1/2cosx2 +c-1/6 (Cos2x)3+c1/2sec u+c-1/6(1-4x)3/2 +c4/3 ln|3x+6|+cx2-8 +c2/5(x+4)5/2 -1/3 (X+4)3/2 +c1/16 (4x-5)4+c2/25(5x2-2)5 +c4/3 sin3x +c-1/10 cos 5x2+c1/8(sin2x)31/2tan3x+c1/9(5+6x)3/2 +c-7/5 ln |5x-8| +c4/33x2+9 +c3(x+4)3/2 -8(x+4)1/2 +c1085/871202/9 ( 17 3/2 – 8 3/2)1/83/20 -3/20 -1296 1/32/3–ln2785/3-7269638/15-7269638/15Ln7-ln5-1/2ln7+1/2ln4 53-52+/- 53x2+c343x3+c±4lnx+c+/- Ce 2x2- 53x2+433343x3-261Y= 3-12lnt+111t-84Y= e-x3+19x-18Y= +/- 45x2+cY= 4x2+cY= 312lnt+cY = +/- Ce32x2Y= 6x2-8Y = - 4x4-15Y= 3-42lnt+24t+192Y=-ln(1/3x2-31x+1)–xcosx+sinx+c2xe2x -1/10 C 2x +c? x4 lnx- 1/16 x4+c1/2x3 sin2x+3/4 x2cos2x-3/4x sin2x-3/8 cos2x+c-5x6 e-x – 30x5e-x – 150x4e-x-600x3e-x-1800x2e-x-3600e-x+c1/3 e 2x sinx+ 2/3 e 2x cosx +c82/81 (-1/3 e-3x cos3x +1/9 e-3x sin3x) +c14/93xsinx+3cosx+c-2xe-x -2e-x +cX4ln3x-1/4x4+c-1/3x4cos3x+4/9x3sin3x+4/9x2cos3x-8/27x sin3x -8/81 cos3x+c1/240 x5e2x -1/96 x4e2x+1/48 x3e2x -1/32 x2e2x +1/32xe2x -1/64 e2x+c4/17 ex sin4x + 1/17 ex cos4x +c-4/17 e4x cosx+ 4/17 e4xsinx64ln4-4ln2-15Y= 2 e3xY= -4e3x2-327138.018 years.230700Y= -3 e5xY=6e2x2-8933 bees85.476 hours.322600MULTIPLE CHOICECABBDDCEABEXTENDED RESPONSEY= e 2ln|x| = |x|2 382713-1427A) P(t) = 88 +/- Ce –kt B) 559 C) 800Y= (5/16x4+75/4x+401/2)2/5 ................
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