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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.
| 101. |
The solution of dy = cos x `(2 - y " cosec " x) dx` , where ` y = sqrt(2) , " when " x = pi//4 ` isA. ` y = sin x +1/2 "cosec"x`B. ` y = tan (x//2)+cot(x//2)`C. `y = (1//sqrt(2))sec (x//2)+sqrt(2)cos (x//2)`D. None of the above |
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Answer» Correct Answer - b Given , `(dy)/(dx) = 2 cos x - y cos x "cosec " x` ` rArr (dy)/(dx) + y cot x = 2 cos x ` which is linear differential equation . ` :. IF = e^(int cot x dx ) = e ^(log (sin x) ) = sin x ` ` :. " Required Solution is " y * sin x = int 2 cos x sin x dx +c` ` rArr y sin x = int sin 2 x dx +C` , ` rArr y sin x = (- cos 2 x)/2 +C ` Given at ` x = pi/4 , y = sqrt(2)` ` :. sqrt(2) sin . pi/4 = (- cos 2(pi //4))/2 +C` ` rArr C = 1` ` :. y sin x = -1/2 cos 2 x +1` ` rArr y = -1/2 * (cos 2x)/(sinx) + " cosec " x ` ` rArr y = 1/(2sin x ) (1-2 sin^(2)x) + " cosec " x` ` rArr y = 1/2 " cosec " x + sin x ` |
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| 102. |
Solution of the differential equation `1+(dy)/(dx)+cosec(x+y)=0` isA. `x-sin(x+y)=c`B. `x+sin(x+y)=c`C. `x-sin(x+y)=c`D. `x+sin(x+y)=c` |
| Answer» Correct Answer - D | |
| 103. |
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)=cx^(2)`D. `x^(2)+y^(2)=cx^(2)` |
| Answer» Correct Answer - B | |
| 104. |
Solution of the differential equation `dr+(2rcot theta+sin2theta)d theta=0` isA. `rsin^(2) theta+(sin^(4) theta)/(4)=c`B. `rsin^(2) theta-(sin^(4) theta)/(4)=c`C. `rsin^(2) theta+(sin^(4) theta)/(2)=c`D. `rsin^(2) theta-(sin^(4) theta)/(2)=c` |
| Answer» Correct Answer - C | |
| 105. |
Solution of the differential equation `(dy)/(dx)=(x-y)/(x+y)` isA. `x^(2)+2xy+y^(2)=c`B. `x^(2)+2xy-y^(2)=c`C. `x^(2)-2xy+y^(2)=c`D. `x^(2)-2xy-y^(2)=c` |
| Answer» Correct Answer - D | |
| 106. |
Solution of the differential equation `(dy)/(dx)+(x-2y)/(2x-y)=0` isA. `(x+y)^(3)=c(y-x)`B. `(x+y)^(3)=c(x-y)`C. `(x-y)^(3)=c(x+y)`D. `(x-y)^(3)=c(x+y)^(2)` |
| Answer» Correct Answer - A | |
| 107. |
Solution of the differential equation `y^(2)dx+(xy+x^(2))dy=0` isA. `xy^(2)=c^(2)(x+2y)`B. `xy^(3)=c^(2)(x+2y)`C. `xy^(2)=c^(2)(x-2y)`D. `xy^(3)=c^(2)(x-2y)` |
| Answer» Correct Answer - A | |
| 108. |
Solution of the differential equation`(9x+5y)dy+(15x+11y)dx=0` isA. `(x+y)^(2)(3x+y)^(3)=c`B. `(x+y)^(3)(3x+y)^(2)=c`C. `(x+y)^(2)(3x+y)^(2)=c`D. `(x+y)^(3)(3x+y)^(3)=c` |
| Answer» Correct Answer - A | |
| 109. |
Solution of differential equation `log((dy)/(dx))=x+y` isA. `e^(x)+e^(y)=c`B. `e^(x)+e^(-y)=c`C. `e^(-x)+e^(y)=c`D. `e^(-x)+e^(-y)=c` |
| Answer» Correct Answer - B | |
| 110. |
Water is dropped at the rate of `2m^(2)//s` into a cone of semivertical angel of `45^(@)`. The rate at which periphery of water surface changes when height of water in the cone is 2 m, isA. 2 m/sB. 1m/sC. 3 m/sD. 4 m/s |
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Answer» Correct Answer - a Given that, `(dV)/(dt) =2` ` rArr d/(dt)(1/3 pir^(3)) = 2 rArr pi r^(2) (dr)/(dt) = 2` ` rArr (dr)/(dt) = 2/(pir^(2))` ` rArr d/(dt) (2pir) = 4/(r^(2))" "` …(i) when H = 2m and r = 2m , then ` (dp)/(dt) = 4/4 = 1m//s ` [ where p =` 2 pi r` = perimeter ] |
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| 111. |
A curve passes through the point (0,1) and the gradient at (x,y) on it is `y(xy-1)`. The equation of the curve isA. `y(x-1)=1`B. `y(x+1)=1`C. `x(y+1)=1`D. `x(y-1)=1` |
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Answer» Correct Answer - a We have , `(dy)/(dx) = y (xy -1)` ` rArr dy = xy^(2)dx - y dx` ` rArr y dx + dy = xy^(2) dx rArr (y" "dx+dy)/(y^(2)) = xdx` ` rArr (ye^(-x) dx + e^(-x) dy)/(y^(2)) = xe^(-x) dx` ` :. -d ((e^(-x))/y) = xe^(-x) dx` On integrating , we get ` - (e^(-x)/y ) = - xe^(-x) - e^(-x)-C` ` rArr 1/y = x+1 + Ce^(x) " " ` ...(i) Since , it passes through (0,1) Therefore , `1 = 1+C rArr C = 0 ` On putting C = 0 in Eq . (i) , we get ` 1/y = x+1 rArr y (x+1) =1` |
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| 112. |
The equation of a curve passing through `(2,7/2)`and havinggradient `1-1/(x^2)`at `(x , y)`is(a)`( b ) (c) y=( d ) x^(( e )2( f ))( g )+x+1( h )`(i)(b) `( j ) (k) x y=( l ) x^(( m )2( n ))( o )+x+1( p )`(q)(c)`( d ) (e) x y=x+1( f )`(g) (d) None of theseA. `y = x^(2) +x+1`B. `xy=x^(2)+x+1`C. `xy=x+1`D. None of these |
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Answer» Correct Answer - b We have, `(dy)/(dx) = 1 - 1/(x^(2))` On integrating both sides , we get ` y = x + 1/x +C` This passes through `(2,7/2)` ` :. 7/2 = 2+ 1/2 + C rArr C = 1` Thus, the equation of the curve is ` y = x + 1/x +1 or xy = x^(2) + x +1` |
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| 113. |
The solution of the differential equation `(1-x^2)(1-y)dx=x y(1+y)dy` isA. `log|x(1-y)^(2)|+(y^(2)-x^(2))/(2)+2y=c`B. `log|x(1-y)^(2)|+(y^(2)+x^(2))/(2)-2y=c`C. `log|x(1-y)^(2)|-(y^(2)+x^(2))/(2)+2y=c`D. `log|x(1-y)^(2)|-(y^(2)+x^(2))/(2)-2y=c` |
| Answer» Correct Answer - A | |
| 114. |
Solution of the differential equation `(dy)/(dx)=(1+y^(2))/(1+x^(2))` isA. `tan^(-1)y-tan^(-1)x=c`B. `tan^(-1)y+tan^(-1)x=c`C. `(tan^(-1)x)+(tan^(-1)y)=0`D. `tan^(-1)y=ctan^(-1)x=0` |
| Answer» Correct Answer - A | |
| 115. |
Solve the differential equation `"dy=cos x(2-y cosec x)dx"` given that `y=2, "when x" d=(pi)/(2)`A. y=sinx+cosxB. `y=tan((x)/(2))+cot((x)/(2))`C. `y=(1)/(sqrt2)sec((x)/(2))+sqrt2cos((x)/(2))`D. y=sinx+cosx |
| Answer» Correct Answer - A | |
| 116. |
Solution of the differential equation `(dy)/(dx)=x^(2)y+y` isA. `log|y|=(1)/(2)x^(2)-x+c`B. `log|y|=(1)/(2)x^(2)+x+c`C. `log|y|=(1)/(3)x^(2)-x+c`D. `log|y|=(1)/(3)x^(2)+x+c` |
| Answer» Correct Answer - D | |
| 117. |
The solution of differential equation ` cos x dy = y (sin x - y ) dx, 0 lt x lt pi //2 ` isA. `sec x = (tan x + C)y`B. `y sec x = tan x +C`C. ` y tan x = sec x +C`D. ` tan x = (sec x + C)y` |
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Answer» Correct Answer - a Since , ` cos x dy = y sinx dx - y^(2)dx` ` rArr 1/(y^(2))(dy)/(dx) -1/y = (dz)/(dx)` ` rArr (dz)/(dx) +(tanx) z = - sec x` This is a linear differential equuation . Therefore `IF = e^(int tanx dx) = e^(log(secx)) = sec x` Hence, the solution is ` z* (sec x) = int - sec x * sec x dx +C_(1)` ` rArr-1/y secx = - tan x + C_(1)` ` rArr sec x = y(tan x +C) " " [ C = -C_(1)]` |
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| 118. |
Solution of the differential equation `(dy)/(dx)=(y(1+x))/(x(y-1))` isA. `log|xy|+x+y=c`B. `log|xy|-x+y=c`C. `log|xy|+x-y=c`D. `log|xy|-x-y=c` |
| Answer» Correct Answer - C | |
| 119. |
Order and degree of the differential equation `((d^(2)y)/(dx^(2)))^(2)+((dy)/(dx))^(3)=e^(x)` respectively areA. 2 and 1B. 2 and 2C. 1 and 3D. 2 and 3 |
| Answer» Correct Answer - B | |
| 120. |
Solution of differential equation `(dt)/(dx)=(xlogx)/(t)` isA. `x=e^(-ct)`B. `x=e^(ct)`C. `log|x|=e^(-ct)`D. `log|x|=e^(ct)` |
| Answer» Correct Answer - B | |
| 121. |
Degree and order of the differential equation `(d^(2)y)/(dx^(2)) = ((dy)/(dx))^(2)` are respectivelyA. 2 and 2B. 1 and 1C. 2 and 1D. 1 and 1 |
| Answer» Correct Answer - C | |
| 122. |
The equation of one of the curves whose slope of tangent at any pointis equal to `y+2x`is`y=2(e^x+x-1)``y=2(e^x-x-1)``y=2(e^x-x+1)``y=2(e^x+x+1)`(5) `y=e^x-x-1`A. `y = 2(e^(x)+x-1)`B. `y=2(e^(x)-x-1)`C. `y = 2(e^(x)-x+1)`D. ` y = 2 (e^(x)+x+1)` |
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Answer» Correct Answer - b Given , `(dy)/(dx) = y + 2x` ` rArr (dy)/(dx) -y = 2x` This is linear differential equation `:. IF = e^((int)-1//dx)= e^(-x)` ` :. ` Solution of the differential equation is ` y * e^(-x) int xe^(-x) dx=2 (-xe^(-x)-e^(-x))+C` [ integration by parts ] ` rArr y = 2e^(x) (-xe^(-x) - e^(-x)) +Ce^(x)` ` rArr y = -2 x - 2 + Ce^(x)` For C =2 we get ` y = 2 (e^(x) -x-1)` ` y = 2(e^(x) - x - 1)` |
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| 123. |
The solution of the differential equationn `(d theta)/(dt)=-k(theta-theta_(0))` where k is constant, is . . . .A. `theta=theta_(0)+ce^(-kt)`B. `theta=theta_(0)+ce^(kt)`C. `theta=theta_(0)+ke^(-kt)`D. `theta=theta_(0)+ke^(kt)` |
| Answer» Correct Answer - A | |
| 124. |
Order and degree of the differential equation `(d^(2)x)/(dt^(2))+((dx)/(dt))^(2)+7=0` respectively areA. 2 and 2B. 1 and 1C. 2 and 1D. 1 and 2 |
| Answer» Correct Answer - C | |
| 125. |
Order and degree of the differential equation `(d^(2)y)/(dx^(2))+x(dy)/(dx)+y=2sinx` respectively areA. 1 and 2B. 1 and 1C. 2 and 1D. 2 and 2 |
| Answer» Correct Answer - C | |
| 126. |
Solution of the differtial equation `y-x(dy)/(dx)=0` isA. xy=0B. y=cxC. xy=0D. y=x |
| Answer» Correct Answer - B | |
| 127. |
The order and degree of the differetnial equation `sqrt(sin x) (dx+dy) = sqrt(cos x )(dx-dy)` areA. (1,2)B. (2,2)C. (1,1)D. (2,1) |
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Answer» Correct Answer - c Given , `sqrt(sin x)(1+(dy)/(dx)) = sqrt(cosx ) (1- (dy)/(dx))` ` rArr (dy)/(dx) = (sqrt(cosx)- sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))` ` :. ` Order =1 , degree =1 |
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| 128. |
The general solution of the differential equation `(dy)/(dx)+(1+cos2y)/(1-cos2x)=0` is given byA. `tan y cot x = C`B. ` tan y - cot x = C`C. `tan x - cot y = C`D. ` tan x + cot y = C` |
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Answer» Correct Answer - b We have , `(dy)/(dx) + (1+cos2y)/(1-cos2x)=0` ` (dy)/(dx) = -(1+cos2y)/(1-cos2y) =-(2cos^(2)y)/(2 sin^(2)x)` ` rArr int sec^(2) y dy = - int "cosec"^(2) x dx` ` rArr tan y = cot x +C` |
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| 129. |
Order and degree of the differential equation `|{:(x^(3),,y^(2),,3),(2x^(2),,3y(dy)/(dx),,0),(5x,,2(y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)),,0):}|` =0 respectively areA. 1 and 1B. 1 and 2C. 2 and 2D. 2 and 1 |
| Answer» Correct Answer - D | |
| 130. |
`y=(sin^(-1)x)^(2)+c` is a solution of the differential equationA. `(1-x^(2))(d^(2)y)/(dx^(2))+x(dy)/(dx)+2=0`B. `(1-x^(2))(d^(2)y)/(dx^(2))+x(dy)/(dx)-2=0`C. `(1-x^(2))(d^(2)y)/(dx^(2))+x(dy)/(dx)+2=0`D. `(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)+2=0` |
| Answer» Correct Answer - D | |
| 131. |
y=logx+c is a solution of the differential equationA. `(d^(2)y)/(dx^(2))-(dy)/(dx)=0`B. `(d^(2)y)/(dx^(2))+(dy)/(dx)=0`C. `(d^(2)y)/(dx^(2))-(dy)/(dx)=0`D. `(d^(2)y)/(dx^(2))+(dy)/(dx)=0` |
| Answer» Correct Answer - D | |
| 132. |
From the differential equation by eliminating A and B in `Ax^(2)+By^(2)=1`A. `xy(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)+y(dy)/(dx)=0`B. `xy(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)-y(dy)/(dx)=0`C. `xy(d^(2)y)/(dx^(2))-((dy)/(dx))^(2)+y(dy)/(dx)=0`D. `xy(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)-y(dy)/(dx)=0` |
| Answer» Correct Answer - B | |
| 133. |
Write the order of the differential equation whosesolution is `y=acosx+b s in x+c e^(-x)dot`A. 3B. 1C. 2D. 4 |
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Answer» Correct Answer - a In the given equation there are three parameters , so its differential equation is third order differential equation . |
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| 134. |
If sinx is the integrating facor (IF) of the linear differetnial equation ` (dy)/(dx)+Py =Q` , then P isA. log sin xB. cos xC. tan xD. cot x |
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Answer» Correct Answer - d We know that , IF of `(dy)/(dx) + Py = Q` is ` IF = e^(int Pdx)` ` sin x = e^(int p dx)` On differentiating both sides , we get ` cos x = e^(int pdx) P` ` cos x = sin P rArr P = cot x` |
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| 135. |
The solution of `(dy)/(dx) = (ax + h)/(by + k)` represents a parabola whenA. a=0,b=0B. a=1,b=2C. a=0,`b!=0`D. a=2 , b = 1 |
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Answer» Correct Answer - c Given ,` (dy)/(dx) = (ax+h)/(by+k) ` ` rArr int (by +k ) dy = int (ax + h ) dx` ` rArr (by^(2))/2 + ky = (ax^(2))/2 + hx +C` Thus , above equation represents a parabola , if `a = 0 andb!= 0` or ` b=0 and a!= 0` |
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| 136. |
If m and n are order and degree of the differential equation `((d^2y)/(dx^2))^5+(4((d^2y)/(dx^2))^3)/((d^3y)/(dx^3))+(d^3y)/(dx^3)=x^2-1` (A) `m=3, n=1` (B) `m=3, n=3` (C) `m=3, n=2` (D) `m=3, n=5`A. m=3,n=3B. m=3,n=2C. m=3,n=5D. m=3,n=1 |
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Answer» Correct Answer - b The given differential equation can be rewritten as ` ((d^(2)y)/(dx^(2)))* (d^(3)y)/(dx^(3))+4((d^(2)y)/(dx^(2)))^(2) +((d^(3)y)/(dx^(3)))= (x^(2)-1)(d^(3)y)/(dx^(3))` ` rArr ` Order (m) = 3 and degree (n) =2 |
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| 137. |
Verify the solution problems: Show that `y = e^-x + ax +b` is solution of the differential equation `e^x d^y / dx^2 = 1`A. `(d^(2)y)/(dx^(2))=e^(-x)`B. `(d^(2)y)/(dx^(2))=e^(x)`C. `(d^(2)y)/(dx^(2))=-e^(-x)`D. `(d^(2)y)/(dx^(2))=-e^(x)` |
| Answer» Correct Answer - A | |
| 138. |
The differential equation for `y=e^(x)(a+bx)` isA. `(d^(2)y)/(dx^(2))+2(dy)/(dx)+y=0`B. `(d^(2)y)/(dx^(2))+2(dy)/(dx)-y=0`C. `(d^(2)y)/(dx^(2))-2(dy)/(dx)+y=0`D. `(d^(2)y)/(dx^(2))-2(dy)/(dx)-y=0` |
| Answer» Correct Answer - C | |
| 139. |
Order and degree of the differential equation `(d^(2)y)/(dx^(2))=root(3)(1+((dy)/(dx))^(3))` respectively areA. 2 and 2B. 2 and 1C. `2 and (1)/(2)`D. 2 and 3 |
| Answer» Correct Answer - D | |
| 140. |
The equation of the curve through the point (1,1) and whose slope is `(2ay)/(x(y-a))` isA. ` y^(a).x^(2a)=e^(y-1)`B. ` y^(a).x^(2a)=e^(y)`C. ` y^(2a).x^(a)=e^(y-1)`D. ` y^(a).x^(a)=e^(y)` |
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Answer» Correct Answer - a We have , slope `(dy)/(dx) = (2ay)/(x(y-a)) rArr (y-a)/y dy = (2a)/x dx` On integrating both sides , we get ` y - a log |y| = 2a log | x| + log C` ` rArr y^(a) * x^(2a) = Ce^(y)` This passes through (1,1) therefore 1 = Ce `rArr C = 1/e` So , the equation of the curve is ` y^(a) * x^(2a) = e^(y-1)` |
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| 141. |
Degree and order of the differential equation `(d^(2)y)/(dx^(2)) = ((dy)/(dx))^(2)` are respectivelyA. 1,2B. 2,1C. 2,2D. 1,1 |
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Answer» Correct Answer - a The given differential eqution is ` (d^(2)y)/(dx^(2)) = ((dy)/(dx))^2` The degree and order of this equation are 1 and 2 respectively . |
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| 142. |
The order and degree of the differential equation ` (d^(2)y)/(dx^(2))=root(3)(1-((dy)/(dx))^(4))` are respectivelyA. 2,3B. 3,2C. 2,4D. 2,2 |
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Answer» Correct Answer - a Given differential equation is ` (d^(2)y)/(dx^(2)) = root3(1-((dy)/(dx))^(4)) rArr ((d^(2)y)/(dx^(2)) ) = 1 - ((dy)/dx)^(4)` ` :. ` Order = 2 , degree = 3 |
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| 143. |
The differential equation obtained by eliminating a and b from `y = ae^(bx)` isA. `y(d^(2)y)/(dx^(2))+(dy)/(dx)=0`B. ` y (d^(2)y)/(dx^(2))-(dy)/(dx)=0`C. `y(d^(2)y)/(dx^(2))-((dy)/(dx))^(2)=0`D. ` y (d^(2)y)/(dx^(2))+((dy)/(dx))^(2)=0` |
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Answer» Correct Answer - c The given equation is ` y=ae^(bx)" "` …(i) On differentiating w.r.t. x, we get `(dy)/(dx)= abe^(bx)` Again , on differentiating w.r.t x, we get ` (d^(2)y)/(dx^(2)) = ab^(2)e^(bx) rArr ae^(bx) (d^(2)y)/(dx^(2)) = a^(2)b^(2)e^(2bx)` ` rArr y (d^(2)y)/(dx^(2)) = (dy/(dx))^(2) rArr y (d^(2)y)/(dx^(2)) - (dy/dx)^(2)=0` which is required differential equation |
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| 144. |
If `y=e^(ax)sinbx,` then `(d^(2)y)/(dx^(2))-2a(dy)/(dx)+a^(2)y=`A. `-a^(2)y`B. `-b^(2)y`C. `-ay`D. `-by` |
| Answer» Correct Answer - B | |
| 145. |
Order and degree of the differential equation `((d^(2)y)/(dx^(2)))^(5)+ (4 ((d^(2)y)/(dx^(2)))^(3))/((d^(3)y)/(dx^(3))) + (d^(3)y)/(dx^(3)) = x^(2) - 1` respectively areA. 3 and 2B. 3 and 1C. 3 and 5D. 3 and 3 |
| Answer» Correct Answer - A | |
| 146. |
The degree of the differential equation ` x = 1 ((dy)/(dx)) + 1/(2!) ((dy)/(dx))^(2) + 1/(3!) ((dy)/(dx))^(3)+…..`A. 3B. 2C. 1D. Not defined |
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Answer» Correct Answer - c ` x = 1 + (dy)/(dx) + 1/(2!) (dy/dx)^(2) +1/(3!) (dy/dx)^(3)+ ....` ` rArr x = e^((dy)/(dx)) rArr (dy)/(dx) = log _(e) x ` ` rArr ` Degree of differential equation is 1 . |
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| 147. |
Which of the following differential equation has the same order and degree?A. `(d^(4)y)/(dx^(4))+8((dy)/(dx))^(6)+5y=e^(x)`B. `5((d^(3)y)/(dx^(3)))^(4)+8(1+(dy)/(dx))^(2)+5y=x^(8)`C. `(1+((dy)/(dx))^(3))^(2/3)=4(d^(3)y)/(dx^(3))`D. `y=x^(2)(dy)/(dx)+sqrt(1+((dy)/(dx))^(2)` |
| Answer» Correct Answer - C | |
| 148. |
The order and degree of the differential equation ` (d^(2)y)/(dx^(2))=root(3)(1-((dy)/(dx))^(4))` are respectivelyA. 1 and 4B. 2 and 1C. 2 and 3D. 2 and 4 |
| Answer» Correct Answer - C | |
| 149. |
Select and write the correct answer from the given alternatives in each of the following sub-questions : The order and degree of the differential equation `[1+((dy)/(dx))^(3)]^(7//3)=7((d^(2)y)/(dx^(2)))` are respectively.A. 3 and 7B. 3 and 2C. 7 and 3D. 2 and 3 |
| Answer» Correct Answer - B | |
| 150. |
The order of the differential equation whose solution is `ae^(x) + be^(2x) + ce^(3x) + d = 0` isA. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - d Since , this equation has 4 arbitrary constants a,b,c and d therefore , order of this differential equation is 4. |
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