Topic: Differential equation

activity:

\fPT] Directions: Solve the given problems. 1. Sketch a family of curves represented by each of the following differential equations: d e (a)=6 m-x Sketch the family of curves given by the equation i- = y and determine the equation of one of these curves which passes through the point [i], ~2}. Find the equivalent DE for all circles whose water is at puinl. (II. D} and thaw sketch of the family of these circles. Find the equivalent DE for all horizontal parabolas whose vertices is at the origin and draw sketch of the family of these parabolas. Find the equivalent DE for all circles of radius 1 unit and draw sketch of the family of these circles. m = b = C 4 = mx + 6 Family of lives : y = CX + 6 20 4' = C ( 1 ) + 0 O y = x + 1 4' = c - y = 2x + 2 0 y = 3x +3 4 = 4 x + 4' y' ( * + 1) - 4 = 0 | + equivalent nc @ y=D / X = ? D=X+1 1 1= 2X + 2 O y= X +1 Q X = D j y = z 0 = 2x 12 2X = -7 * sketch of the family of lines 1 1 =3x + 3 that inter rect @ point ( - 1 0 ) . Q X = b / 4 = 3 @ 420ix = ?-|IT 2 3 5 8 9 X ( x - 4)* = = 40) (4- 10 ) v V(1,k) 2 dy clx = 4X 1 P(1,3) (y - 1)" = = 4p ( x -h) = 4x dx Jay = 4 fx dx y = 4 X +C y = 2 x ' t c - 2 x* = 4 - C Family of paratiberi Z x ' = y + c ( x - 0) 7 = = [ 4 - (-4] 2 O ( x- 0 )' = 4 X NKO ( x - 0) " = 2 (4+ 4 ) X + C V (0 , - 4 ) ( x - 0 ) " = 4 + CV Solving for particular solm , " 0 x 3. ( 2 - 0 ) = 2 + 6 4 = 4+0 Oxy 3 ( x - 0) = 4( x - 0 ) ' = 4 + 2 X' = 0 + 2 X =NZ X1 = 4141 X 1 = - 1.41 2 ( x - 0) = 1 x - 4/ + 5 x = y + 3 ( x - 0 )" = = (4 + 6 ) ( x - 0 )' = = Ly-(-67] * Sketch of the family of upward parabolas where vertius are all located v ( 0 , - 6 ) at the y-axis. @ yzo ; x = ? 4 0= = X' = 0 + 3 P = 0. 125 X = = 15 ( x - ( 4) " = 4p ( 4 -19 ) X, = + 1.73 ( x - 0 ] = 1 + 3 X1 = - 1.43 C O ( x - 0) = 14+() ( x - 0 ) " = + (4 + 6 ) ( x - 0] = = (y + 5 ) V ( o , -s ) 14 p - @ y = D (x - 0) * = ( = )( y + 5 ) X'= D + 5/2 P = 6. 125 X = 11.58 ( x-0) ? = = ly-((3)] X1 = THAN - 1.58