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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Find the general solution of the differential equation `x(dy)/(dx)+2y=x^2(x!=0)`.A. `y=(x^(2)+c)/(4x^(2)`B. `y=(x^(2))/(4)+c`C. `y=(x^(4)+c)/(x^(2)`D. `y=(x^(4)+c)/(4x^(2)` |
| Answer» Correct Answer - D | |
| 2. |
The solution of the differential equation `x+y(dy)/(dx)=2y` is |
| Answer» Correct Answer - B | |
| 3. |
Solution of the differential equation `(dy)/(dx)=(y+sqrt(x^(2)-y^(2)))/(x)` isA. `sin^(-1)((x)/(y))-log|x|=c`B. `sin^(-1)((x)/(y))+log|x|=c`C. `sin^(-1)((y)/(x))-log|x|=c`D. `sin^(-1)((y)/(x))+log|x|=c` |
| Answer» Correct Answer - C | |
| 4. |
Solution of the differential equation `(x^(2)-y^(2))dx-2xy dy=0` isA. `x^(2)-y^(2)=cx`B. `x^(2)+y^(2)=cx`C. `x^(2)-y^(2)=c`D. `x^(2)+y^(2)=c` |
| Answer» Correct Answer - A | |
| 5. |
Show that the differential equation `2xy (dy)/(dx) = x^2+3y^2 ` is homogeneousand solve it.A. `x^(3)+y^(2)=cx^(2)`B. `(x^(2))/(2)+(y^(3))/(3)=y^(2)+c`C. `x^(2)+y^(3)=cx^(2)`D. `x^(2)+y^(2)=cx^(2)` |
| Answer» Correct Answer - D | |
| 6. |
Solution of the differential equation `x^(2)y dx-(x^(3)+y^(3))dy=0` isA. `log|y|=(x^(3))/(3y^(3))+c`B. `log|y|=(-x^(3))/(3y^(3))+c`C. `log|y|=(x^(3))/(y^(3))+c`D. `log|y|=(-x^(3))/(y^(3))+c` |
| Answer» Correct Answer - A | |
| 7. |
Solution of the differential equation `x(dy)/(dx)=y+sqrt(x^(2)+y^(2))`, isA. `y-sqrt(x^(2)+y^(2))=cx^(2)`B. `y+sqrt(x^(2)+y^(2))=cx^(2)`C. `y-sqrt(x^(2)+y^(2))=cx`D. `y+sqrt(x^(2)+y^(2))=cx` |
| Answer» Correct Answer - B | |
| 8. |
Solution of the differential equation `x^(2)y dy+(x^(3)+x^(2)y-2xy^(2)-y^(3))dx=0` isA. `log|(y+x)/(x^(4)(y-x))|=(4x)/(x+xy)+c`B. `log|(y-x)/(x^(4)(y+x))|=(4x)/(x+xy)+c`C. `log|(y+x)/(x^(4)(y-x))|=(4x)/(x+y)+c`D. `log|(y-x)/(x^(4)(y+x))|=(2x)/(x+y)+c` |
| Answer» Correct Answer - D | |
| 9. |
Solution of the differential equation `(x^(2)+2y^(2))dx-xy dy=0,` when y (9)=0 isA. `x^(4)=-81(x^(2)+y^(2))`B. `x^(4)=81(x^(2)+y^(2))`C. `x^(4)=-9(x^(2)+y^(2))`D. `x^(4)=9(x^(2)+y^(2))` |
| Answer» Correct Answer - B | |
| 10. |
The solution of the differential equation ` xy^(2) dy - (x^(3) +y^(3))dx = 0 ` isA. `y^(3)=3x^(3)+c`B. `y^(3)=3x^(3)log|cx|`C. `y^(3)=3x^(3)+log|cx|`D. `y^(3)+3x^(3)=log|cx|` |
| Answer» Correct Answer - B | |
| 11. |
What is the solution of the differential equation `(dy)/(dx)=xy+x+y+1` ?A. `log|1+y|=x-x^(2)+c`B. `log|1+y|=x+x^(2)+c`C. `log|1+y|=x+(x^(2))/(2)+c`D. `log|1+y|=x-(x^(2))/(2)+c` |
| Answer» Correct Answer - C | |
| 12. |
Solution of the differential equation `(xsqrt(x^(2)-y^(2))-y^(2))dx+xy dy=0` isA. `sqrt(x^(2)+y^(2))=log|cx|`B. `sqrt(x^(2)+y^(2))=-log|cx|`C. `sqrt(x^(2)+y^(2))=xlog|cx|`D. `sqrt(x^(2)+y^(2))=-xlog|cx|` |
| Answer» Correct Answer - D | |
| 13. |
Solution of the differential equation `xy^(3)(dy)/(dx)=1-x^(2)+y^(2)-x^(2)y^(2)` isA. `x^(2)+y^(2)+log|x^(2)+x^(2)y^(2)|=c`B. `x^(2)+y^(2)-log|x^(2)+x^(2)y^(2)|=c`C. `x^(2)-y^(2)+log|x^(2)+x^(2)y^(2)|=c`D. `x^(2)-y^(2)-log|x^(2)+x^(2)y^(2)|=c` |
| Answer» Correct Answer - B | |
| 14. |
Solution of the differrential equation `(dy)/(dx)=xsqrt(25-x^(2))` isA. `y-(25-x^(2))^(3/2)=c`B. `y+(25-x^(2))^(3/2)=c`C. `3y-(25-x^(2))^(3/2)=c`D. `3y+(25-x^(2))^(3/2)=c` |
| Answer» Correct Answer - D | |
| 15. |
Solution of the differential equationA. `2y^(2)tan^(-1)x-1=cy^(2)`B. `y^(2)tan^(-1)x-1=cy^(2)`C. `2y^(2)tan^(-1)x+1=cy^(2)`D. `y^(2)tan^(-1)x+1=cy^(2)` |
| Answer» Correct Answer - C | |
| 16. |
If C is an arbitrary constant , then the general solution of the differential equation ` y dx - x dy = xy dx ` is given byA. ` y = Cxe^(-x)`B. ` y = Cye^(-x)`C. ` y + e^(x) = Cx`D. ` ye^(x) = Cx` |
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Answer» Correct Answer - d Given , ` y (1-x) dx = xdy ` ` rArr (1/x -1) dx = 1/y dy ` ` rArr log (x) - x log y - log C` [ integrating ] ` rArr x = log. (xC)/y rArr ye^(x) = xC` |
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| 17. |
The family of curves `y = e^(a sinx) ` where a is an arbitrary constant , is represented by the differential equationA. ` y sin x= C + soin 2x`B. ` y cos x = C +1/2 sin 2x`C. ` y cos x = C - sin 2x`D. ` y cos x = C +1/2 cos 2x` |
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Answer» Correct Answer - d Given , `(dy)/(dx) - y tan x = -2 sin x` ` :. IF = e^( - int tanx dx) cos x ` ` :.` Solution is ` y (cos x ) = int -2 sin 2x dx +C` ` rArr y cos x = (cos 2x)/2 + C` |
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| 18. |
The differential equation for `x^(3)+y^(3)=4ax` isA. `3xy^(2)(dy)/(dx)+2x^(3)+y^(3)=0`B. `3xy^(2)(dy)/(dx)+2x^(3)-y^(3)=0`C. `3xy^(2)(dy)/(dx)-2x^(3)+y^(3)=0`D. `3xy^(2)(dy)/(dx)-2x^(3)-y^(3)=0` |
| Answer» Correct Answer - B | |
| 19. |
The number of arbitraryconstants in the general solution of a differential equationof fourth order are:(A) 0 (B) 2 (C) 3 (D)4A. zeroB. 2C. 3D. 4 |
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Answer» Correct Answer - d We know that the number of arbitary constants in the general solution of a differential equation of order n is equal to its order . Therefore , the number of arbitrary constants in the general equation of fourth order differential equation is four . |
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| 20. |
Consider the equation `(x^2)/(a^2+lambda)+(y^2)/(b^2+lambda)=1,`where a and b are specified constants and `lambda`is an arbitrary parameter. Find a differential equation satisfied by it.A. `(dy)/(dx)=sqrt((1-y^(2))/(1-x^(2)))`B. `(dy)/(dx)=-sqrt((1-y^(2))/(1-x^(2)))`C. `(dy)/(dx)=sqrt((1-x^(2))/(1-y^(2)))`D. `(dy)/(dx)=-sqrt((1-x^(2))/(1-y^(2)))` |
| Answer» Correct Answer - C | |
| 21. |
If a an arbitrary constant, then solution of the differential equation `(dy)/(dx)+sqrt((1-y^(2))/(1-x^(2)))=0` isA. `sin^(-1)x+sin^(-1)y=c`B. `sin^(-1)x-sin^(-1)y=c`C. `sin^(-1)xsin^(-1)y=c`D. `sin^(-1)x=csin^(-1)y` |
| Answer» Correct Answer - A | |
| 22. |
The differential equation whose general solutionis given by `y=(c_1cos(x+c_2)-(c_3e^((-x+c4))+(c_5sinx),`where `c_1,c_2,c_3,c_4,c_5`arearbitrary constants, is(a) `( b ) (c) (d)(( e ) (f) d^(( g )4( h ))( i ) y)/( j )(( k ) d (l) x^(( m )4( n ))( o ))( p ) (q)-( r )(( s ) (t) d^(( u )2( v ))( w ) y)/( x )(( y ) d (z) x^(( a a )2( b b ))( c c ))( d d ) (ee)+y=0( f f )`(gg)(hh)`( i i ) (jj) (kk)(( l l ) (mm) d^(( n n )3( o o ))( p p ) y)/( q q )(( r r ) d (ss) x^(( t t )3( u u ))( v v ))( w w ) (xx)+( y y )(( z z ) (aaa) d^(( b b b )2( c c c ))( d d d ) y)/( e e e )(( f f f ) d (ggg) x^(( h h h )2( i i i ))( j j j ))( k k k ) (lll)+( m m m )(( n n n ) dy)/( o o o )(( p p p ) dx)( q q q ) (rrr)+y=0( s s s )`(ttt)(uuu)`( v v v ) (www) (xxx)(( y y y ) (zzz) d^(( a a a a )5( b b b b ))( c c c c ))/( d d d d )(( e e e e ) d (ffff) x^(( g g g g )5( h h h h ))( i i i i ))( j j j j ) (kkkk)+y=0( l l l l )`(mmmm)(nnnn)`( o o o o ) (pppp) (qqqq)(( r r r r ) (ssss) d^(( t t t t )3( u u u u ))( v v v v ) y)/( w w w w )(( x x x x ) d (yyyy) x^(( z z z z )3( a a a a a ))( b b b b b ))( c c c c c ) (ddddd)-( e e e e e )(( f f f f f ) (ggggg) d^(( h h h h h )2( i i i i i ))( j j j j j ) y)/( k k k k k )(( l l l l l ) d (mmmmm) x^(( n n n n n )2( o o o o o ))( p p p p p ))( q q q q q ) (rrrrr)+( s s s s s )(( t t t t t ) dy)/( u u u u u )(( v v v v v ) dx)( w w w w w ) (xxxxx)-y=0( y y y y y )`(zzzzz)A. 6B. 5C. 4D. 3 |
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Answer» Correct Answer - b Given , `y = c_(1) cos ( x+c_(2))+c_(3) sin (x+c_(4)) +c_(5)e^(x) +c_(6)` ` y = c_(1) [ cos x cos c_(2) - sin x sin c_(2)]` ` + c_(3) [ sin x cos_(4) + cos x sin c_(4)] + c_(5) e^(x) +c_(6)` ` = cos x (c_(1) cos c_(2) + c_(3) sin c_(4))` ` + sin x ( - c_(1) sin c_(2) +c_(3) cos c_(4)) + c_(5)e^(x) +c_(6)` ` = A cos x + B sin x + Ce^(x) + D` where ` A = c_(1) cos c_(2) + c_(3) sin c_(4)` `B = - c_(1) sin c_(2) + c_(3) cos c_(4) , C= c_(5) , D = c_(6)` Hence , order is 4 . |
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| 23. |
The differential equation of all circles whose radius is 5 centre is any point (h,k) isA. `25((d^2y)/(dx^2))+(1+((dy)/(dx))^2)^2 =0`B. `25((d^2y)/(dx^2))-(1+((dy)/(dx))^2)^2 =0`C. `25((d^2y)/(dx^2))+(1+((dy)/(dx))^2)^3 =0`D. `25((d^2y)/(dx^2))-(1+((dy)/(dx))^2)^3 =0` |
| Answer» Correct Answer - D | |
| 24. |
The differential equation for all circles with centre (h,0) and radius r,h and r being arbitary constants isA. `y^(2)-2xy(dy)/(dx)-x^(2)=0`B. `y^(2)-2xy(dy)/(dx)+x^(2)=0`C. `y^(2)+2xy(dy)/(dx)-x^(2)=0`D. `y^(2)+2xy(dy)/(dx)+x^(2)=0` |
| Answer» Correct Answer - A | |
| 25. |
The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given byA. ` y^(2) ((dy)/(dx))^(2)= k^(2) - y^(2)`B. `[ y(dy)/(Dx)]^(2)=k^(2)-y^(2)`C. ` y(dy)/(dx)=k^(2)-y^(2)`D. `[ y(dy)/(dx)]^(2)=k^(2)+y^(2)` |
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Answer» Correct Answer - a The length of normal is given by ,` y sqrt(1+((dy)/(dx))^(2))` ` :. y sqrt (1+((dy)/(dx))^(2))= k ` [ given ] ` rArr y^(2) [ 1+ ((dy)/(dx))^(2) ] = k^(2) rArr y^(2) + y^(2) ((dy)/(dx))^(2)= k^(2)` ` rArr y^(2) ((dy)/(dx))^(2)= k^(2) - y^(2)` |
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| 26. |
Bismath decomposes at a rate proportional to the quanitity of the substance present. If initial mass of substance is 800 mg. and its half life is of 5 days, then the mass of bismath after 30 days isA. 12.5 mgB. 25 mgC. 6.25 mgD. 18.75 mg |
| Answer» Correct Answer - A | |
| 27. |
Solution of differential equation `(x-y^(2)x)dx-(y+x^(2)y)dy=0,ifx=2,y=0,` isA. `(1+x^(2))(1-y^(2))=4`B. `(1+x^(2))(1-y^(2))=5`C. `(1+x^(2))(1-y^(2))=0`D. `(1+x^(2))(1-y^(2))=-5` |
| Answer» Correct Answer - B | |
| 28. |
Solution of differential equation `y-x(dy)/(dx)=y^(2)+(dy)/(dx),` when x=1,y=2, isA. (1+x)(1-y)+y=0B. (1+x)(1-y)-y=0C. (1+x)(1+y)+y=0D. (1+x)(1+y)-y=0 |
| Answer» Correct Answer - A | |
| 29. |
Solution of differential equation `cos""(dy)/(dx)=a,ain R,y(0)=2,` isA. `sin((y+2)/(x))=a`B. `sin((y-2)/(x))=a`C. `cos((y+2)/(x))=a`D. `cos((y-2)/(x))=a` |
| Answer» Correct Answer - D | |
| 30. |
The solution of the differential equation `dy/dx=sin(x+y)+cos(x+y)` is:A. `1+tan(x+y)=ce^(x)`B. `1-tan(x+y)=ce^(x)`C. `1+tan((x+y)/(2))=ce^(x)`D. `1-tan((x+y)/(2))=ce^(x)` |
| Answer» Correct Answer - C | |
| 31. |
Solution of the differential equation `(x(dy)/(dx)-y)sin((y)/(x))=x^(2)e^(x)` isA. `e^(x)+cos((y)/(x))=c`B. `e^(x)-cos((y)/(x))=c`C. `xe^(x)+cos((y)/(x))=c`D. `xe^(x)-cos((y)/(x))=c` |
| Answer» Correct Answer - A | |
| 32. |
Solution of the differential equation `sin^(-1)((dy)/(dx))=x+y` isA. tan( x+y) - sec(x+y)-x=cB. tan(x+y)+sec(x+y)-x=cC. tan(x+y)-sec(x=y)+x=cD. tan(x+y)+sec(x+y)+x=c |
| Answer» Correct Answer - A | |
| 33. |
Solution of the differential equation `(x-y)(1-(dy)/(dx))=e^(x)` isA. `(x-y)^(2)-e^(x)=c`B. `(x-y)^(2)+e^(x)=c`C. `(x-y)^(2)/(2)-e^(x)=c`D. `(x-y)^(2)/(2)+e^(x)=c` |
| Answer» Correct Answer - C | |
| 34. |
Solution of the differential equation `(1+e^(x/y))dx + e^(x/y)(1-x/y)dy=0` isA. `x+ye^(-x/y)=c`B. `x-ye^(-x/y)=c`C. `x+ye^(x/y)=c`D. `x-ye^(x/y)=c` |
| Answer» Correct Answer - C | |
| 35. |
Solution of the differential equation `(x-y)^2(dy/dx)=a^2` isA. `alog|(x-y-a)/(x-y+a)|-y=c`B. `alog|(x-y-a)/(x-y+a)|-y=c`C. `(a)/(2)log|(x-y-a)/(x-y+a)|+y=c`D. `(a)/(2)log|(x-y-a)/(x-y+a)|-y=c` |
| Answer» Correct Answer - D | |
| 36. |
The solution of the differential equation ` (dy)/(dx) = (x-2y+1)/(2x -4y)` isA. `(x-2y)^(2+2x=C`B. `(x-2y)^(2)+x=C`C. `(x-2y)+2x^(2)=C`D. `(x-2y)+x^(2)=C` |
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Answer» Correct Answer - a Given , ` (dy)/(dx) = (x-2y +1)/(2x-4y)` , Put, ` x - 2y = z rArr 1 -2 (dy)/(dx) = (dz)/(dx)` ` :. 1/2 [ -(dz)/(dx) +1] = (z +1)/zx ` ` rArr zdx = -dx` ` rArr (z^(2))/2 = - x +C_(1)` |
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| 37. |
Solve `(1+2e^(x//y))dx+2e^(x//y)(1-x//y)dy=0.`A. `x+2ye^(x/y)=c`B. `x-2ye^(x/y)=c`C. `x+2ye^(y/x)=c`D. `x-2ye^(y/x)=c` |
| Answer» Correct Answer - A | |
| 38. |
If the integrating factor of the differential equation `(dy)/(dx) +P` (x) y = Q (x) is x , then P (x) isA. xB. `x^(2)//2`C. `1//x`D. `1//x^(2)` |
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Answer» Correct Answer - a Given , If = x ` :. e^(int Pdx) = x` ` rArr int Pdx = log x rArr P = d/(dx) log x = 1/x ` |
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| 39. |
`(dy)/(dx) =e^(2y) cos x, " when " x = pi/6, y = 0`A. `e^(-2y)+2sinx=2`B. `e^(-2y)+2sinx=-2`C. `e^(-2y)+sinx=1`D. `e^(-2y)+sinx=-1` |
| Answer» Correct Answer - A | |
| 40. |
The solution of the differential equation `e^(-x) (y+1) dy +(cos^(2) x - sin 2x) y (dx) = 0` subjected to the condition that y = 1 when x = 0 isA. `y+logy+e^(x)cos^(2)x=2`B. `log(y+1)+e^(x)cos^(2)x=1`C. `y+logy=e^(x)cos^(2)x`D. `(y+1)+e^(x)cos^(2)x=2` |
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Answer» Correct Answer - a Given , ` e^(-x) (y+1) dy + (cos^(2) x - sin 2x ) ydx = 0 ` ` rArr (1+1/y ) dy = - e^(x) (cos ^(2)x - sin 2 x) dx ` On integrating both sides , we get ` y + lo y = -e^(x) cos^(2) x + int e^(x) sin 2x dx` ` -int e^(x) sin 2x dx + C` ` rArr y + log y = -e^(x) cos^(2) x + C ` At x = 0 , y = 1 ` 1+ 0 = - e^(0) cos 0 + C rArr C = 2` ` :. "Required solution is " y + log y = = -e^(x) cos^(2) x +2` |
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| 41. |
The solution of the differential equation `(x+y)^(2)(dy)/(dx) = a^(2) ` isA. `y-tan^(-1)((x+y)/(a))=c`B. `y+tan^(-1)((x+y)/(a))=c`C. `y-atan^(-1)((x+y)/(a))=c`D. `y+atan^(-1)((x+y)/(a))=c` |
| Answer» Correct Answer - C | |
| 42. |
If `(dy)/(dx)+y=2e^(2x)`, then y is equal toA. ` Ce^(x)+2/3e^(2x)`B. ` (1-x)e^(-x)+2/3e^(2x)+C`C. ` Ce^(-x)+2/3e^(2x)`D. `e^(-x)+2/3e^(2x)+C` |
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Answer» Correct Answer - a Given , ` (dy)/(dx) +y = 2e^(2x)` This is linear differential equation ` :. IF = e^(int 1 dx) = e^(x)` ` :. " Required solution is "` ` ye^(x) = 2 int e^(2x) e^(x) dx + C = 2/3 e^(3x) +C` ` rArr y = 2/3 e^(2x) + Ce^(-x)` |
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| 43. |
`3e^(x) tan y dx + (1+e^(x)) sec^(2) dy =0 , " when" x = 0 and y = pi`A. `(1+e^(x))^(3)tany=0`B. `(1+e^(x))^(3)tany=4`C. `(1+e^(x))^(3)coty=0`D. `(1+e^(x))^(3)coty=4` |
| Answer» Correct Answer - A | |
| 44. |
The solution of `(x+y+1)(dy)/(dx) = 1` isA. ` y = (x+2)+Ce^(x)`B. `y=-(x+2)+Ce^(x)`C. `x=-(y+2)+Ce^(y)`D. `x=(y+2)^(2)+Ce^(y)` |
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Answer» Correct Answer - c Given , `(dx)/(dy) = x+y+1` ` rArr (dx)/(dy) - x = y +1` ` :. IF = e^(int-1dy) = e^(-y)` Its solution is given by ` x . IF = int IF xx Q dy + C` ` rArr x * e^(-y) = int (y+1)e^(-y) + int e^(-y) dy + C` ` rArr xe^(-y) = - (y+1) e^(-y) -e^(-y) +C` ` rArr xe^(-y) = - (y+1) e^(-y) -e^(-y) +C` ` rArr x = - (y+2) +Ce^(y)` |
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| 45. |
The solution of the differential equation `(dy)/(dx)=(4x+y+1)^(2)`, isA. 4x= y+1 =tan (2x+2c)B. 4x + y+1 = cot (2x+2c)C. 4x + y+1 = 2tan (2x+2c)D. 4x + y+1 =2 cot (2x+2c) |
| Answer» Correct Answer - C | |
| 46. |
The solution of the differential equation `e^(-x) (y+1) dy +(cos^(2) x - sin 2x) y (dx) = 0` subjected to the condition that y = 1 when x = 0 isA. `y=log|y|+e^(x)cos^(2)x=2`B. `log|y+1|+e^(x)cos^(2)x=1`C. `y+log|y|=e^(x)cos^(2)x`D. `log|y+1|+e^(x)cos^(2)x=2` |
| Answer» Correct Answer - A | |
| 47. |
The solution of the differential equation `(dy)/(dx) +y/x = sin x` isA. ` x(y+cos x)= sin x+C`B. ` x (y-cos x) = sin x+C`C. ` x (y cos x)= sin x+C`D. ` x (y- cos x)= cos x+C` |
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Answer» Correct Answer - a Given , `(dy)/(dx) + y/x= sin x , :. IF = e^(int 1/x dx) = x` ` :. " Solution is " y * x = int x sin x dx + C` ` rArr xy -x cos x + sin x +C` ` rArr ( y + cos x) = sin x + C` |
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| 48. |
The solution of `(dy)/(dx) + y tan x = sec x ` isA. ` y sec x = tan x +C`B. ` y tan x = sec x +C`C. ` tan x = y tan x +C`D. ` x sec x = y tan y +C` |
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Answer» Correct Answer - a Given , `(dy)/(dx) + y tan x = sec x ` ` :. IF = e^(intP dx)= e^(int tanxdx) = sec x ` ` :. " Solution is ", y sec x = int sec^(2)x dx +C` ` rArr y sec x = tan x +C` |
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| 49. |
For solving ` dy/dx = 4x +y +1`, suitable substitution isA. y+4x+1=uB. y=4x+uC. y=4xD. y=ux |
| Answer» Correct Answer - A | |
| 50. |
The integrating factor of the differential equation ` (dy)/(dx) + 1/x* y = 3x ` isA. xB. In xC. 0D. `oo` |
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Answer» Correct Answer - c Given , `(dy)/(dx) +1/x* y = 3x` ` :. IF = e^(int 1/x dx) = e^(log x) = x` |
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