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                                   Math112(Calculus 2) July 2013 – Instructor: Ivan Matic
                                                Differential Equations
                                        1                                          ′
                  1. Show that y = x− x is a solution to the differential equation xy +y = 2x.
                                                                               x2
                  2. Show that every member of the family of functions y =Ce 2 is a solution of the differential
                     equation y′ = xy.
                                                                        2
                  3.   (a) What can you say about a solution of y′ = −y just looking at the differential equation
                           (without solving it)?                       3
                       (b) Verify that all members of the family y =  3   are solutions to the equation from the
                           part (a).                                 x+c
                                                                            y2
                       (c) Canyouthinkofasolutiontotheequationy′ =− 3 thatisnotamemberofthefamily
                           from (b).
                                                                            y2
                       (d) Find a solution to the initial value problem y′ = − 3 , y(0) = 0.5.
                  4. Solve the following equations:
                            ′         ′                  ′         y   ′
                       (a) y =y; (b) y =−y; (c) cosy·y =x; (d) e ·y =x;
                       (e) cosy·y′·x+siny=0;(f) siny·y′·x=cosy;
                            y  ′      y
                       (g) e ·y ·x+e =sinx.
                                        Separable Differential Equations
                  1. Solve the differential equations:
                            ′           ′          ′    2       ′            2 ′
                       (a) y =3y; (b) y =xy; (c) y = x y; (d) y = xcosy+x y ;
                            ′
                       (e) y =cosx+cosx·y2;(f) y′ =(1−y2)(1−x2).
                           x
                  2. Solve the initial value problems:
                       (a) dy =y2+1,y(1)=0;(b) dy = ycosx,y(0)=1;
                           dx                        dx    1+y2
                           dy     y                               3y  ′
                       (c) dt =te , y(1) = 0; (d) xcosx = (2y+e )y , y(0) = 0.
                  3. Find the equation of the curve that satisfies dy = 4x3y and whose y-intercept is 7.
                                                                dx
                  4. Find the equation of the curve that passes through the point (1,1) and whose slope at (x,y)
                        y2
                     is x2.
                                                               1
                                                      ′     p 2
                  5. Solve the initial value problem y = 2x   1−y ,y(0)=0.
                  6. Solve the differential equations:
                            ′      y    y       ′                        ′       x       y
                       (a) y =cosx+x;(b)xy =(lny−lnx)y+y;(c)y = xey/x+y+x;
                                x3+y2x                  y4          y            y2
                            ′                 ′                            ′
                       (d) y =    x2y   ; (e) y = x4+x3y+x2y2 + x; (f) y = x2+y2.
                                           Linear Differential Equations
                  1. Solve the differential equations using the product rule:
                            3x ′      3x           ′                      ′
                       (a) e y +3ye =x;(b)xy +y=xlnx;(c)sinx·y +cosx·y=tanx;
                            ′         −3x        ′   1             ′
                       (d) y +3y=e       ·x; (e) y + xy = lnx; (f) y +cotx·y = secx;
                            ′               ′  1              ′
                       (g) y +3y=x;(h)y +xy=sinx;(i)y +cotx·y=sinx;
                            ′               ′             ′
                       (j) y +4y=3;(k)y −y=x;(l)y +sinx·y=sin(2x).
                  2. Solve the differential equations:
                            ′          x      ′               ′         2
                       (a) y +2y=2e ;(b)y =x+5y;(c)xy −2y=x ;
                            x          ′           du
                       (d) e +xy=xy;(e)(1+t)dt +u=1+t,t >0;
                            ′
                       (f) y =x+y,y(0)=2;
                           dv            2 t2
                       (g) dt −2tv=3t e ;
                       (h) xy′ = y+x2sinx; y(π) =0.
                  3. If f is any differentiable function such that f′(x)+xf(x) is bounded, calculate the limit
                       lim f(x).
                      x→+∞
                                    Differential Equations of Second Order
                  1. Solve the differential equations:
                            ′′         ′′         ′′          ′′
                       (a) y =0; (b) y =8; (c) y =x; (d) y =sinx;
                            ′         ′′         ′           ′′
                       (e) y =y; (f) y =y; (g) y = 3y; (h) y = 9y.
                  2. Solve the equations:
                            ′′    ′                ′′    ′
                       (a) y −8y +12y=0;(b)y −3y +2y=0;
                            ′′   ′               ′′    ′
                       (c) y −y −6y=0;(d)y −2y +y=0;
                            ′′    ′              ′′   ′
                       (e) y −4y +4y=0;(f)y +y −6y=0;
                                                               2
                    ′′  ′          ′′
                (g) y +6y +9y=0;(g)y −y=0.
             3. Solve the equations:
                    ′′  ′               ′
                (a) y +2y +2y=0,y(0)=2,y(0)=1;
                (b) 2y′′+5y′+3y=0,y(0)=3,y′(0)=−4;
                    ′′  ′               ′
                (c) y −2y +5y=0,y(π)=0,y(π)=−2;
                    ′′  ′
                (d) y +8y +41y=0;
                (e) 4y′′+y=0, y(0)=3, y(π)=4;
                    ′′  ′
                 (f) y −6y +25y=0,y(0)=1,y(π)=2.
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