17%20-%20Differential%20Equations

17%20-%20Differential%20Equations - PROBLEMS(1 17...

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PROBLEMS 17 - DIFFERENTIAL EQUATIONS Page 1 ( 1 ) Prove that y = A e x + B e - 2x + x 2 + x is a solution of 2 2 2 2x 3 2y dx dy dx y d - - = + , ( A, B are arbitrary constants ) ( 2 ) Solve ( x 2 + yx 2 ) dy + ( y 2 - xy 2 ) dx = 0. + = y 1 x 1 cx y log : Ans ( 3 ) Solve y 2 y x e x e dx dy - - + = and find the particular solution subject to initial condition x = 1, y = 1. + = + + = 3 1 x e e c, 3 x e e : Ans 3 x y 3 x y - ( 4 ) Solve xydx - ( x 2 + y 2 ) dy = 0. = 2 2y 2 x e c y : Ans ( 5 ) Solve x dx dy = y + x x y cos 2 . = cx log x y tan : Ans l l ( 6 ) Solve y dx - x dy + 2 2 y x - dx = 0. ( x > 0 ). = l l cx log x y sin : Ans 1 - ( 7 ) Solve ( 2x - 3y + 4 ) dx + ( 3x - 2y + 1 ) dy = 0. [ Ans: ( x + y - 3 ) 5 = c ( y - x - 1 ) ] ( 8 ) Solve ( 2x + y ) dx - ( 4x + 2y - 1 ) dy = 0. [ Ans: log l 10x + 5y - 2 l = 10y - 5x + c ] ( 9 ) Solve dx dy + 2y = x 2 . [ Ans: 4y = 2x 2 - 2x + 1 + c e 2x ]
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PROBLEMS 17 - DIFFERENTIAL EQUATIONS Page 2 ( 10 ) y - intercept of tangent at any point ( x, y ) on a curve is 2xy 2 . Find the equation of the curve. [ Ans: x - x 2 y = cy ] ( 11 ) The population of a country is doubled in 50 years. If the rate of increase of the population is proportional to the population, how many years will it take to become three times the original population ? [ Ans: 79 years ] ( 12 ) Prove that 2yy” = ( y’ ) 2 has solution x = at + b, y = bt 2 , ( a, b are arbitrary constants and t is a parameter ). ( 13 ) Prove that the differential equation of a family of circles having centres on Y - axis and touching X - axis is ( x 2 - y 2 ) dx dy = 2xy. Find
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17%20-%20Differential%20Equations - PROBLEMS(1 17...

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