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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.
| 301. |
The differential equation obtained by eliminating a and b from `y = ae^(bx)` isA. `y_(2)=y_(1)+y`B. `y_(2)^(2)=yy_(1)`C. `y_(1)^(2)=yy_(2)`D. `y^(2)=y_(1)y_(2)` |
| Answer» Correct Answer - C | |
| 302. |
`y=ax^(2)+bx+c` is the general solution of the D.E. A)`y_(1)=0` B)`y_(2)=0` C)`y_(3)=x` D)`y_(3)=0`A. `y_(1)=0`B. `y_(2)=0`C. `y_(3)=x`D. `y_(3)=0` |
| Answer» Correct Answer - D | |
| 303. |
Find the differential equation from the equation `(x-h)^(2)+(y-k)^(2)=a^(2)` by eliminating `h` and `k`. |
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Answer» `(x-h)^(2)+(y-k)^(2)=a^(2)`………..`(1)` `implies 2(x-h)+2(y-k)(dy)/(dx)=0` `implies(x-h)=-(y-k)*(dy)/(dx)` `implies1=-[(y-k)*(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)]` `implies y-k=-(1+((dy)/(dx))^(2))/((d^(2)y)/(dx^(2)))` `(x-h)=(1+((dy)/(dx))^(2))/((d^(2)y)/(dx^(2)))*(dy)/(dx)` From eq. `(1)` `([1+((dy)/(dx))^(2)]^(2))/(((d^(2)y)/(dx^(2)))^(2))((dy)/(dx))^(2)+([1+((dy)/(dx^(2)))])/(((d^(2)y)/(dx^(2)))^(2))=a^(2)` `implies[1+((dy)/(dx))^(2)]^(2)*[((dy)/(dx))^(2)+1]=a^(2)*((d^(2)x)/(dx))^(2)` `implies[1+((dy)/(dx))^(2)]^(3)=a^(2)*((d^(2)x)/(dx))^(2)` |
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| 304. |
Find the differential equations corresponding to `v=(A)/(r )+B`. |
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Answer» `v=(A)/(r )+B` …………..`(1)` `implies (dv)/(dr)=-(A)/(r^(2))implies-A=r^(2)*(dv)/(dr)` `implies0=[r^(2)*(d^(2)v)/(dr^(2))+(dv)/(dr)*2r]` `implies r*(d^(2)v)/(dr^(2))+2(dv)/(dr)=0`. which is the required differential equation. |
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| 305. |
Find the differential equation for the equation `x^(2)+y^(2)-2ax=0`. |
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Answer» `x^(2)+y^(2)-2ax=0` ………..`(1)` Differntitate with respect to `x` `2x+2y(dy)/(dx)-2a=0` `implies a=x+y(dy)/(dx)` From eq. `(1)` `x^(2)+y^(2)-2x(x+y(dy)/(dx))=0` `implies y^(2)-x^(2)-2xy(dy)/(dx)=0` whichis the required differential equations. |
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| 306. |
Form differential equation for `(x)/(a)+(y)/(b)=1`A. `y_(3)=0`B. `(x)/(y_(1))+(y)/(y_(2))=1`C. `y_(1)y_(2)=1`D. `y_(2)=0` |
| Answer» Correct Answer - D | |
| 307. |
Form differential equation for `Ax+By+C=0`A. `y_(3)=0`B. `y_(2)=0`C. `y_(3)x+y_(2)y+y_(1)=0`D. `x+yy_(1)+y_(2)=0` |
| Answer» Correct Answer - B | |
| 308. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. `(d^(3)y)/(dx^(3))=1`B. `(d^(3)y)/(dx^(3))`C. `(d^(3)y)/(dx^(3))=0`D. none of these |
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Answer» Correct Answer - C The equation of the parabola having their axes parallel to y-axis is given by `y=ax^(2)+bx+c` `rArr" "(dy)/(dx)=2ax+brArr(d^(2)y)/(dx^(2))=2arArr(d^(3)y)/(dx^(3))=0` This is the required differential equation. |
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| 309. |
The solution of the differential equation `(dy)/(dx)+1=e^(x+y)`, isA. `(x+y).e^(x+y)=0`B. `(x+c).e^(x+y)=0`C. `(x-c).e^(x+y)=1`D. `(x-c).e^(x+y)+1=0` |
| Answer» Correct Answer - D | |
| 310. |
If `(dp)/(dy) = 3^(cos y) sin y`, then P is equal toA. sin y + CB. `3^(cos y) + C`C. `(-3^(cos y))/(ln 3) + C`D. `3^(sin y) + c` |
| Answer» Correct Answer - C | |
| 311. |
`y=a cos x+b sin x+x sin x` is a solution of the D.E. A)`y_(2)+y=x cos x` B)`y_(2)+2y=2cos x` C)`y_(2)+y=2cos x` D)`y_(2)-y=2cos x`A. `y_(2)+y=x cos x`B. `y_(2)+2y=2cos x`C. `y_(2)+y=2cos x`D. `y_(2)-y=2cos x` |
| Answer» Correct Answer - C | |
| 312. |
D.E. of all circules of a given radius a and variable centre (h, k) isA. `(1+y_(1)^(2))^(3)+a^(2)(y_(2))^(2)=0`B. `(1+y_(1)^(2))^(2)+a^(2)(y_(2))^(3)=0`C. `(1+y_(1)^(2))^(3)=a^(2)(y_(2))^(2)`D. `2xdx+2ydt=a^(2)` |
| Answer» Correct Answer - C | |
| 313. |
The differenital equation of all circles whose centres are at the origin isA. `ydx-xdy=0`B. `xdx+ydy=0`C. `ydx+xdy=0`D. `xdx-ydy=0` |
| Answer» Correct Answer - B | |
| 314. |
If a = order and b = degree of the D.E. `y_(2)=(1+y_(1)^(2))^(3//2),` then `a+b=…` A)1 B)3 C)5 D)4A. 1B. 3C. 5D. 4 |
| Answer» Correct Answer - D | |
| 315. |
The differential equation of all parabolas whoseaxis are parallel to the y-axis is(a)`( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )`(s)(b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)`(kk)(c)`( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )`(ii)(d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)`(ggg)A. `y_(2) = 2y_(1) + x`B. `y_(3) = 2y_(1)`C. `y_(2)^(3) = y_(1)`D. `y_(3) = 0` |
| Answer» Correct Answer - D | |
| 316. |
Find the differential equation of all parabolas whose axis are parallel to the x-axis.A. 2B. 3C. 1D. none of these |
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Answer» Correct Answer - B The general equation of all parabolas whose axis of symmetry is parallel to x axis is `x=ay^(2)+by+c`, where a, b, c are arbitrary constants. As there are arbittrary constants. So, the requried differential equation is of order 3. |
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| 317. |
The order of the differential equation whose general solution is given by `y=(C_(1)+C_(2))sin(x+C_(3))-C_(4)e^(x+(C_(5))`, isA. 5B. 4C. 3D. 2 |
| Answer» Correct Answer - C | |
| 318. |
The order of the differential equation whose general solution is `y = c_(1) cos 2x + c_(2) cos^(2) x + c_(3) sin^(2) x + c_(4)` isA. 2B. 4C. 3D. None of these |
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Answer» Correct Answer - A `y = c_(1) cos 2x + c_(2)cos^(2)x + c_(3) sin^(2) x + c_(4)` `=c_(1) cos 2x + (c_(2))/(2)(1+cos 2x)+(c_(3))/(2)(1-cos 2x)+c_(4)` `=((c_(2))/(2)+(c_(3))/(2)+c_(4))+(c_(1)+(c_(2))/(2)-(c_(3))/(2))cos 2x` `=A + B cos 2x` That means there are two independent parameter. Thus, the order of differential equation will be 2. |
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| 319. |
The order of the differential equation whose general solution is `y = (C_(1) + C_(2)) cos (x + C_(3)) - C_(4)e^(x^(4))` where `C_(1), C_(2), C_(3)` and `C_(4)` are arbitrary isA. 2B. 3C. 4D. 5 |
| Answer» Correct Answer - B | |
| 320. |
Order of the differential equation `(d^(2)y)/(dx^(2))+5(dy)/(dx)+intydx=x^(3)` isA. 2B. 3C. 1D. 4 |
| Answer» Correct Answer - B | |
| 321. |
The solution of equation `(2y-1)dx-(2x+3)dy=0` isA. `(2x-1)/(2y+3)=k`B. `(2y+1)/(2x-3)=k`C. `(2x+3)/(2y-1)=k`D. `(2x-1)/(2y-1)=k` |
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Answer» Given that, `" "(2y-1)dx-(2x+3)dy=0` `rArr" "(2y-1)dx=(2x+3)dy` `rArr" "(dx)/(2x+3)=(dy)/(2y-1)` On integrating both sides, we get `" "(1)/(2) log(2x+3)=(1)/(2)log(2y-1)+logC` `rArr" "(1)/(2)[log*(2x+3)-log(2y-1)]=logC` `rArr" "(1)/(2)log((2x+3)/(2y-1))=logC` `" "((2x+3)/(2y-1))^(1//2)=C` `rArr" "(2x+3)/(2y-1)=C^(2)` `rArr" "(2x+3)/(2y-1)=k`, where `K=C^(2)` |
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| 322. |
The degree of the differential equation `y_(3)^(2//3)+2+3y_(2)+y_(1)=0`, isA. 1B. 2C. 3D. none of these |
| Answer» Correct Answer - B | |
| 323. |
D.E., having the solution `y=c_(1)+c_(2)e^(3x)`, is A)`y_(2)=3y` B)`y_(2)=3y_(1)` C)`y_(3)+3y_(1)=0` D)`y_(2)+3y=0`A. `y_(2)=3y`B. `y_(2)=3y_(1)`C. `y_(3)+3y_(1)=0`D. `y_(2)+3y=0` |
| Answer» Correct Answer - B | |
| 324. |
Derivative of `y=ae^(bx+c)`A. `y_(1)^(2)=yy_(2)`B. `y_(3)=yy_(1)^(2)`C. `y_(3)=y^(2)y_(1)`D. `y^(3)=y_(1)y_(2)y_(3)` |
| Answer» Correct Answer - A | |
| 325. |
`root3((dy)/(dx)sqrt((d^(3)y)/(dx^(3))))=5` . Find order and degree of differential equation.A. 3, 3B. 3, 1C. 3, 6D. 3, 2 |
| Answer» Correct Answer - B | |
| 326. |
The differential equation for which `y=a cos x+b sin x` is a solution isA. `(d^(2)y)/(dx^(2))+y=0`B. `(d^(2)y)/(dx^(2))-y=0`C. `(d^(2)y)/(dx^(2))+(a+b)y=0`D. `(d^(2)y)/(dx^(2))+(a-b)y=0` |
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Answer» Given that, `y=acosx+bsinx` On differentiating both sides w.r.t. x, we get `" "(dy)/(dx)=-asinx+bcosx` Again, differentiating w.r.t. x, we get `" "(d^(2)y)/(dx^(2))=-asinx+bcosx` `rArr" "(d^(2)y)/(dx^(2))=-y` `rArr" "(d^(2)y)/(dx^(2))+y=0` |
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| 327. |
The degree and the order of the differential `y=x((dy)/(dx))^2+((dx)/(dy))^2` are respectivelyA. 1, 1B. 2, 1C. 4, 1D. 1, 4 |
| Answer» Correct Answer - C | |
| 328. |
`|(x,y),(1,x(dy)/(dx)+(dy)/(dx))|=0` . Find order and degree of differential equation.A. 1, 1B. 1, 0C. `1, -1`D. `1, (1)/(2)` |
| Answer» Correct Answer - A | |
| 329. |
Derivative of `y=ae^(x)+b`A. `y_(1)=y`B. `y_(2)=y_(1)`C. `y=y_(1)y_(2)`D. `y_(2)+y_(1)=0` |
| Answer» Correct Answer - B | |
| 330. |
Derivative of `y=ae^(-x)+b`A. `y_(2)=y_(1)`B. `y+y_(1)+y_(2)=0`C. `y_(2)=yy_(1)`D. `y_(1)+y_(2)=0` |
| Answer» Correct Answer - D | |
| 331. |
The differential equation satisfying all the curves `y = ae^(2x) + be^(-3x)`, where a and b are arbitrary constants, isA. `y_(2)=y_(1)-6y=0`B. `y_(2)-y_(1)+6y=0`C. `y_(2)+y_(1)-6y=0`D. `y_(1)-y_(2)+6y=0` |
| Answer» Correct Answer - C | |
| 332. |
The solution of `(dy)/(dx)+y=e^(-x), y(0)=0 "is"`A. `y=e^(-x)(x-1)`B. `y=xe^(x)`C. `y=xe^(-x)+1`D. `y=xe^(-x)` |
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Answer» Given that, `" "(dy)/(dx)+y=e^(-x)` which is a linear differential equation Here, `P=1 and Q=e^(-x)` `" "IF=e^(intdx)=e^(x)` The general solution is `" "y*e^(x)=inte^(-x)*e^(x)dx+C` `rArr" "ye^(x)=intdx+C` `rArr" "ye^(x)=x+C" "`...(i) When x=0 and y=0 then, 0=0+C `rArr` C=0 Eq. (i) becomes `y*e^(x)=xrArr=xe^(-x)` |
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| 333. |
Derivative of `y=e^(x)(c_(1)+c_(2)x)`A. `(y_(1))^(2)-2y_(1)+1=0`B. `y_(2)-2y_(1)+y=0`C. `y_(2)+2y_(1)-y=0`D. `y_(2)=2y_(1)+y` |
| Answer» Correct Answer - B | |
| 334. |
The order and degree of differential equation: `((d^(3)y)/(dx^(3)))^(2)-3(d^(2)y)/(dx^(2))+2((dy)/(dx))^(4)=y^(4)"are"`A. 1,4B. 3,4C. 2,4D. 3,2 |
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Answer» Given that, `((d^(3)y)/(dx^(3)))^(2)-3(d^(2)y)/(dx^(2))+2((dy)/(dx))^(4)=y^(4)` `therefore "Order"=3` and degree=2 |
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| 335. |
The order and degree of differential equation: `[1+((dy)/(dx))^(2)]=(d^(2)y)/(dx^(2)"are"`A. `2,(3)/(2)`B. 2,3C. 2,1D. 3,4 |
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Answer» Given that, `((d^(3)y)/(dx^(3)))^(2)-3(d^(2)y)/(dx^(2))+2((dy)/(dx))^(4)=y^(4)` `therefore "Order"=3` and degree=2 `[(1+(dy)/(dx))^(2)]=(d^(2)y)/(dx^(3))` Order=2 and degree=1 |
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| 336. |
Which of the following is a general solution of `(d^(2)y)/(dx^(2))-2(dy)/(dx)+y=0`A. `y=(Ax+B)e^(x)`B. `y=(Ax+B)e^(-x)`C. `y=Ax^(x)+Be^(-x)`D. `y=Acos x+Bsinx` |
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Answer» Given that, `(d^(2)y)/(dx^(2))-2(dy)/(dx)+y=0` `D^(2)-2Dy+y=0` `"where" D=(d)/(dx)` `(D^(2)-2D+1)y=0` The auxiliary equation is `m^(2)-2m+1=0 `(m-1)^(2)+1=0` `(m-1)^(2)=0 Rightarrow m=1,1` Since, the roots are real and equal `CF=(Ax+B)e^(x) Rightarrow y=(Ax+B)e^(x)` Since, if roots of Auxiliary equation are real and equal say (m), then `CF=(C_(1)x+C_(2))e^(mx)]` |
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| 337. |
Find the particular solution of the differential equation `(1+e^(2x))dy+(1+y^2)e^x dx=0,`given that `y=1`when `x=0.` |
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Answer» `(1+e^(2x))dy+(1+y^(2))e^(x)dx=0` or `(dy)/(1+y^(3))+(e^(x)dx)/(1+e^(2x))=0` Integrating both sides, we get `int(dy)/(1+y^(2))+int(e^(x)dx)/(1+e^(2x))=C` `Now, tan^(-1)y+tan^(-1)(e^(x))=C`…………..(1) `therefore tan^(-1)+tan^(-1)=C` or `C=pi/2` Substituting `C=pi/2`, in equation(1), we get `tan^(-1)y+tan^(-1)(e^(x))=pi/2` This is the required solution of the given differential equation. |
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| 338. |
From the differential equation of the family curves having equation `y=(sin^(-1)x)^(2)+Acos^(-1)x+B`. |
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Answer» `y=(sin^(-1)x)^(2)+Acos^(-1)x+B` `=(sin^(-1)x)^(2)-Asin^(-1)sin^(-1)x+(piA)/2+B` Differentiating w.r.t.x, we get `(dy)/(dx) = (2sin^(-1)x)/sqrt(1-x^(2))-A/sqrt(1-x^(2))` `rArr (1-x^(2))((dy)/(dx))^(2)=4(sin^(-1)x)^(2)-4Asin^(-1)x+A^(2)` `=4y-2piA-4B+A^(2)` Differentiating again, w.r.t, x we get `2(1-x^(2))(dy)/(dx)(d^(2)y)/(dx^(2))-2x((dy)/(dx))^(2)=4(dy)/(dx)` `rArr (1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)=2`, which is required differential equation. |
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| 339. |
What is the order of the differential equation whose general solution is `y=c_(1)cos2x+c_(2)sin^(2)x+c_(3)cos^(2)x+c_(4)e^(2x)+c_(5)e^(2x+c6)`? |
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Answer» `y=c_(1)cos2x+c_(2)2sin^(2)x+c_(3)cos^(2)x+c_(4)e^(2x)+c_(5)e^(2x+c6)` `c_(1)cos2x+c_(2)(1-cos2x)/(2)+c_(3)(cos2x+1)/(2) + c_(4)e^(2x)+c_(5)e^(c6)e^(2x)` `=(c_(1)-c_(2)/2+c_(3)/2)cos2x+(c_(2)/2+c_(3)2)+(c_(4)+c_(5)e^(c6))e^(2x)` `=lambda_(1)cos2x+lambda_(2)e^(2x)+lambda_(3)` So, number of artibatry constants in the equation is 3. Therefore, order of the differential equation will be 3. |
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| 340. |
Find the differential equation of the family ofcurves `y=A e^(2x)+B e^(-2x)`, where A and B are arbitrary constants. |
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Answer» Correct Answer - `(d^(2)y)/(dx^(2))=4y` `y=Ae^(2x)+Be^(-2x)` `therefore (dy)/(dx)=2Ae^(2x)-2Be^(-2x)` or `(d^(2)y)/(dx^(2))=4Ae^(2x)+4Be^(-2x)` = `4(Ae^(2x)+Be^(-2x))` `=4y`, which is the required differential equation. |
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| 341. |
The differential equation of family of curves of `y^(2)=4a(x+a)`isA. `y^(2)=4(dy)/(dx)((x+dy)/(dx))`B. `2y (dy)/(dx)=4a`C. `(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)=0`D. `2x(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)-y=0` |
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Answer» Given that, `y^(2)=4a(x+a)…(i)` On differentiating both sides w.r.t. x, we get `2y(dy)/(dx)=4a Rightarrow 2y(dy)/(dx)=4a` `Rightarrow y(dy)/(dx)=2a Rightarrow a=(1)/(2)y(dy)/(dx).........(ii)` On putting the value of a form Eq. (ii) in Eq. (i) we get `y^(2)=2y(dy)/(dx) (x+(1)/(s)y(dy)/(dx))` `Rightarrow y^(2)=2xy(dy)/(dx)+y^(2)((dy)/(dx))^(2)` `Rightarrow 2x(dy)/(dx)+y((dy)/(dx))^(2)-y=0` |
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| 342. |
Form the differential equation from the followingprimitives where constants are arbitrary: `y^2=4a x`A. `yy_(1)=2x`B. `y_(1)=2xy`C. `y=2xy_(1)`D. `xyy_(1)=2` |
| Answer» Correct Answer - C | |
| 343. |
The solution of the differential equation `(dy)/(dx) = 1/(xy[x^(2)siny^(2)+1])` isA. `x^(2)(cosy^(2)-siny^(2)-2Ce^(-y^(2)))=2`B. `y^(2)(cosx^(2)-siny^(2)-2Ce^(-y^(2)))=4C`C. None of theseD. a system of circles |
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Answer» Correct Answer - A `(dy)/(dx) = 1/(xy[x^(2)siny^(2)+1])` or `1/x^(3)(dx)/(dy) -1/x^(2)y=ysiny^(2)` Putting `-1//x^(2)=u`, we get `(du)/(dy)+2uy=2ysiny^(2)`. I.F. `=e^(y^(2))` Thus, solution is `ue^(y^(3))=int2ysiny^(2)e^(y^(2))dy+C` `=int(sint)e^(t)dt+C` `=1/2e^(y^(3))(siny^(2)-cosy^(2))+c` or `2u=(siny^(2)-cosy^(2))+2Ce^(-y^(2))` or `2=x^(2)[cosy^(2)-siny^(2)-2Ce^(-y^(2))]` |
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| 344. |
Solve the differential equation `(dy)/(dx) = y + int_(0)^(1)y(x)dx`, given that the value of y is 1, when x = 0. |
| Answer» `y = (2e^(x) - e+1)/(3-e)` | |
| 345. |
The solution of `(dy)/(dx)=(x^2+y^2+1)/(2x y)`satisfying `y(1)=1`is given by(a) a system of parabolas(b) a system of circles(c)`( d ) (e) (f) y^(( g )2( h ))( i )=x(( j ) (k)1+x (l))-1( m )`(n)(d) `( o ) (p) (q) (r)(( s ) (t) x-2( u ))^(( v )2( w ))( x )+( y ) (z)(( a a ) (bb) y-3( c c ))^(( d d )2( e e ))( f f )=5( g g )`(hh)A. a system of parabolasB. a system of circlesC. `y^(2)=x(1+x)-1`D. `(x-2)^(2)+(y-3)^(2)=5` |
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Answer» Correct Answer - C Rewritting the given equation is `2xy(dy)/(dx) -y^(2)=1+x^(2)` or `2y(dy)/(dx)-1/xy^(2)=1/x+x` Putting `y^(2)=u`, we have `(du)/(dx) -1/xu=1/x+x` I.F. `=e^(-int1/xdx)=1/x` Thus, solution is `u1/x=int(1/x^(2)+1)dx=-1/x+x+C` or `y^(2)=(x^(2)-1)+Cx` Since `y(1)=1`, we get C=1. Hence, `y^(2)=x(1+x)-1` which represents a system of hyperbola. |
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| 346. |
If the curve satisfying `(xy^(4) + y)dx - xdy = 0` passes through (1,1) then the value `-41(y(2))^(3)` is _______ |
| Answer» Correct Answer - 32 | |
| 347. |
The solution of `(dy)/(dx)=(x^2+y^2+1)/(2x y)`satisfying `y(1)=1`is given by(a) a system of parabolas(b) a system of circles(c)`( d ) (e) (f) y^(( g )2( h ))( i )=x(( j ) (k)1+x (l))-1( m )`(n)(d) `( o ) (p) (q) (r)(( s ) (t) x-2( u ))^(( v )2( w ))( x )+( y ) (z)(( a a ) (bb) y-3( c c ))^(( d d )2( e e ))( f f )=5( g g )`(hh)A. a hyperbolaB. a circleC. `y^(2)=x(1+x)-10`D. `(x-2)^(2)+(y-3)^(2)=5` |
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Answer» Correct Answer - A We have, `(dy)/(dx)=(x^(2)+y^(2)+1)/(2xy)` `rArr" "2xy dy=(x^(2)+y^(2)+1)dx` `rArr" "2xydy-y^(2)dx=(x^(2)+1)dx` `rArr" "xd(y^(2))-y^(2)dx=(x^(2)+1)dx` `rArr" "(xd(y^(2))-y^(2)dx)/(x^(2))=(1+(1)/(x^(2)))dx` `rArr" "d((y^(2))/(x))=d(x-(1)/(x))` On integrating, we get `(y^(2))/(x)=x-(1)/(x)+C` `rArr" "y^(2)=x^(2)-1+Cx rArr y^(2)=(x+(C)/(2))^(2)-1-(C^(2))/(4)` Clearly, it represents a hyperbola. |
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| 348. |
Let I be the purchase value of an equipment and V(t) be the value after ithas been used for t years. The value V(t) depreciates at a rate given bydifferential equation `(d V(t)/(dt)=-k(T-t)`, where `k"">""0`is a constant and T is thetotal life in years of the equipment. Then the scrap value V(T) of theequipment is :(1) `T^2-1/k`(2) `I-(k T^2)/2`(3) `I-(k(T-t)^2)/2`(4) `e^(-k T)`A. `e^(-kT)`B. `T^(2)-I/k`C. `I-(kT^(2))/2`D. `I-(k(T-t)^(2))/(2)` |
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Answer» Correct Answer - C Since total life is T, scrap value is V(T). We have `(dV)(t)=-k(T-t)dt` `rArr int_(I)^(V(t))dV(t) = int_(t=0)^(T)-k(T-t)dt` `rArr V(T)-I = k[((T-t)^(2))/(2)]_(0)^(T)` `rArr V(T)-I = -k[t^(2)/2]` `rArr V(T)=I-(kT^(2))/2` |
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| 349. |
Let `f:R to R` be a differentiable function with `f(0)=0`. If `y=f(x)` satisfies the differential equation `(dy)/(dx)=(2+5y)(5y-2)`, then the value of `lim_(x to oo) f(x)` is………………. |
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Answer» Correct Answer - D `(dy)/(dx) = (5y+2)(5y-2)` `rArr 1/25int(dy)/(y^(2)-(2/5)^(2))=intdx` `=1/25.5/4log_(e)|(y-2)/(y+2/5)|=x+c` `=1/20 log_(e)|(5y-2)/(5y+2)|=x+c` Given that `f(0)=y(0)=0`. `therefore c=0` Hence, `|(2-5y)/(2+5y)|=e^(20x)` `therefore lim_(x to infty) |(2-5y)/(2+5y)|=lim_(x to -infty)e^(20x)` `rArr lim_(x to -infty)|(2-5y)/(2+5y)|=0` `lim_(x to -infty)y=2/5=0.4` |
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| 350. |
If `y=y(x)`satisfies the differential equation `8sqrt(x)(sqrt(9+sqrt(x)))dy=(sqrt(4+sqrt(9+sqrt(x))))^(-1)dx ,x >0a n dy(0)=sqrt(7,)`then `y(256)=`16 (b)80 (c) 3(d) 9A. 3B. 9C. 16D. 80 |
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Answer» Correct Answer - A We have, `dy=(dx)/(8sqrt(x)(sqrt(9+sqrt(x))(sqrt(4+sqrt(9+sqrt(x))))))` Let `4+sqrt(9+sqrt(x))=t` `rAr 1(2sqrt(9+sqrt(x))).1/(2sqrt(x))dx=dt` `rArr dy=(dt)/(2sqrt(t))` `rArr 2dy=1/sqrt(t)dt` `rArr 2y=2sqrt(t)+c` `rArr 2y=2sqrt(4+sqrt(9+sqrt(x)))` Given, `y(0) = sqrt(7) rArr c=0` `therefore y=sqrt(4+sqrt(9+sqrt(x)))` `therefore y(256)=3` |
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