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The general solution of the DE ( y - x...

The general solution of the DE ` ( y - x ) dy + (x + y ) dx = 0 ` is

A

` tan ^(-1)""(y)/(x) = C sqrt (x ^(2) + y ^(2))`

B

` e ^(tan ^(-1) ( y//x)) = C sqrt (x ^(2) + y ^(2))`

C

`tan ^(-1) ((y)/(x))= x ^(2) + y ^(2) + C `

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
` (dy)/(dx) = ( x+ y )/( x - y ) `, which is clearly homogeneous.
Put ` y= vx and (dy)/(dx) = v + x ( dv )/(dx) rArr int ((1 - v ))/(( 1+ v ^(2))) dv = int (1)/(x) dx `
`rArr int (dv)/((1 + v ^(2))) - (1)/(2) int (2v)/((1+ v^(2))) dv = int (1)/(x) dx `
`rArr tan ^(-1) v - (1)/(2) log (1 + v ^(2)) = log x + log C rArr tan ^(-1) v = log (Cx sqrt (1 + v ^(2)))`
` rArr tan ^(-1) "" ( y )/(x) = log (C sqrt ( x ^(2) + y ^(2))) rArr e ^( tan ^(-1) y//x) = C sqrt (x ^(2) + y ^(2))`.
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RS AGGARWAL-LINEAR DIFFERENTIAL EQUATIONS -Objective Questions
  1. The solution of the DE (dy)/(dx) = e ^( x + y ) + x ^(2)* e ^(y...

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  2. The solution of the DE (dy )/(dx) + sqrt ((1 -y ^(2))/( 1- x ^(2))...

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  3. The solution of the DE (dy)/(dx) = (1 - cos x )/( 1+ cos x ) is

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  4. The solution of the DE (dy)/(dx) = (- 2 xy )/( ( x ^(2) + 1)) is

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  5. The solution of the DE cos x (1 + cosy )dx - sin y (1+ sin x ) dy=0 ...

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  6. The solution of the DE x cos y dy = (x e ^(x) log x + e ^(x)) dx is

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  7. The solution of the DE (dy)/(dx) + y log y cot x = 0 is

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  8. The general solution of the DE (1+ x ^(2)) dy - xy dx = 0 is

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  9. The solution of xsqrt(1+y^2)dx+ysqrt(1+x^2)dy=0

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  10. The solution of the equation log ((dy)/(dx))=a x+b y is (a) ( b ) (...

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  11. The general solution of the DE (dy)/(dx) = (sqrt (1 - x ^(2))) (sqrt...

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  12. The general solution of the DE (dy)/(dx) = ( y^(2)- x ^(2))/( 2xy ...

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  13. The general solution of the DE x ^(2) (dy)/(dx) = x ^(2) + xy + y ^...

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  14. The general solution of the DE x (dy)/(dx) = y + x tan "" (y)/(x) ...

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  15. The general solution of the DE 2x y dy + (x ^(2) - y ^(2)) dx = 0...

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  16. The general solution of the DE ( y - x ) dy + (x + y ) dx = 0 i...

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  17. The general solution of the DE (dy)/(dx) = (y)/(x) + sin ""(y)/(x) ...

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  18. The general solution of the DE (dy)/(dx) + y tanx = sec x is

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  19. The general solution of the DE (dy)/(dx) + y cot x = 2 cos x is

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  20. The general solution of the DE (dy)/(dx) + (y)/(x) = x ^(2) is

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