Solve the following :
Question: A body slipping on a rough horizontal plane moves with a deceleration of $4.0 \mathrm{~m} / \mathrm{s}^{2}$. What is the coefficient of kinetic friction between the block and the plane? Solution: Vertical equilibrium $\mathrm{N}=\mathrm{mg}$ For horizontal; ${ }^{\text {Net }}={ }_{\text {ma }}$ $0-f f=m a$ $-\mu N=m a$ $-\mu m g=m a$ $-\mu g=\alpha$ $-\mu g=(-4)$ $\mu=0.4 \mathrm{~m} / \mathrm{s}^{2}$...
Read More →The differential equation for y = A cos αx + B sin αx,
Question: The differential equation fory= A cos x+ B sin x, where A and B are arbitrary constants is (A) $\frac{d^{2} y}{d x^{2}}-\alpha^{2} y=0$ (B) $\frac{d^{2} y}{d x^{2}}+\alpha^{2} y=0$ (C) $\frac{d^{2} y}{d x^{2}}+\alpha y=0$ (D) $\frac{d^{2} y}{d x^{2}}-\alpha y=0$ Solution: Correct option is (B). Given equation isy= A cos ax+ B sin ax Differentiating both sides w.r.t. x, we have...
Read More →If y = e–x (A cos x + B sin x),
Question: Ify=ex(A cosx+ B sinx), thenyis a solution of (A) $\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}=0$ (B) $\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$ (C) $\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}+2 y=0$ (D) $\frac{d^{2} y}{d x^{2}}+2 y=0$ Solution: Correct option is (C). Given equation,y=ex(A cosx+ B sinx) Differentiating on both the sides, w.r.t. x, we get...
Read More →The order and degree of the differential equation
Question: The order and degree of the differential equationrespectively, are $\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x^{\frac{1}{5}}=0$ (A) 2 and not defined (B) 2 and 2 (C) 2 and 3 (D) 3 and 3 Solution: Correct option is (A) 2 and not defined. Given differential equation is As the degree of dy/dx is in fraction its undefined and the degree is 2....
Read More →The degree of the differential equation
Question: The degree of the differential equation (A) 4 (B) 3/2 (C) not defined (D) 2 Solution: Correct option is (D) 2. Given differential equation is...
Read More →The degree of the differential equation
Question: The degree of the differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=x \sin \left(\frac{d y}{d x}\right)$ is: (A) 1 (B) 2 (C) 3 (D) not defined Solution: Correct option is (D) not defined. Since the value of sin (dy/dx) on expansion will be in increasing power of dy/dx, the degree of the given differential equation is not defined....
Read More →Solve the following
Question: Solve : $x \frac{d y}{d x}=y(\log y-\log x+1)$ Solution:...
Read More →Find the equation of a curve passing through the point (1, 1).
Question: Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P (x,y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB. Solution: Lets take P(x , y) be any point on the curve and AB be the tangent to the given curve at P. Also given, P is the mid-point of AB So, the coordinates of A and B are (2x, 0) and (0, 2y) respectively....
Read More →Find the equation of a curve passing through origin if the slope
Question: Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x,y) is equal to the square of the difference of the abscissa and ordinate of the point. Solution: We know that, The slope of the tangent of the curve = dy/dx And the difference between the abscissa and ordinate = x y So, as given in the question we have dy/dx = (x y)2 Taking, x y = v...
Read More →Find the equation of the curve through the point (1, 0)
Question: Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x,y) is (y 1)/ (x2+ x) Solution: Given that the slope of the tangent to the curve at (x, y) is As, the line is passing through the point (1, 0), then (0 1) (1 + 1) = c(1) ⇒ c = 2 Thus, the required solution is (y 1) (x + 1) = 2x....
Read More →Find the equation of a curve passing through
Question: Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any point (x,y) is (x2+ y2)/ 2xy. Solution: Given that the slope of tangent to a curve at (x, y) is (x2+ y2)/ 2xy. Its a homogeneous differential function...
Read More →Find the general solution of
Question: Find the general solution ofdy/dx 3y = sin 2x. Solution: Given equation,dy/dx 3y = sin 2x Its a first order linear differential equation Here P = -3 and Q = sin 2x...
Read More →Solve: dy/dx = cos(x + y) + sin (x + y).
Question: Solve:dy/dx= cos(x+y) + sin (x+y). [Hint: Substitutex+y=z] Solution:...
Read More →Find the general solution of
Question: Find the general solution of (1 + tany) (dxdy) + 2xdy= 0. Solution: Given, (1 + tany) (dxdy) + 2xdy= 0...
Read More →The value of
Question: Solve : $y+\frac{d}{d x}(x y)=x(\sin x+\log x)$ Solution: Given differential equation,...
Read More →Find the differential equation of system
Question: Find the differential equation of system of concentric circles with centre (1, 2). Solution: The family of concentric circles with centre (1, 2) and radius r is given by (x 1)2+ (y 2)2= r2 Differentiating both sides w.r.t, x we get...
Read More →Solve the differential equation
Question: Solve the differential equation (1 +y2) tan1x dx+ 2y(1 +x2)dy= 0. Solution: Given differential equation, (1 +y2) tan1x dx+ 2y(1 +x2)dy= 0 2y (1 + x2) dy = -(1 + y2)tan-1x dx Thus, the above equation is the required solution....
Read More →Form the differential equation by eliminating A and B
Question: Form the differential equation by eliminating A and B in Ax2+ By2= 1. Solution: Given, Ax2+ By2= 1 Differentiating w.r.t. x, we get...
Read More →Solve the differential equation dy = cos x (2 – y cosec x)
Question: Solve the differential equationdy= cosx(2 ycosecx)dxgiven thaty= 2 when x = /2. Solution: Given differential equation,dy= cosx(2 ycosecx)dx...
Read More →Solve : 2 (y + 3) – xy dy/dx = 0,
Question: Solve : 2 (y+ 3) xy dy/dx =0, given that y(1) = -2. Solution: Given differential equation, 2 (y+ 3) xy dy/dx =0...
Read More →Solve : (x + y) (dx – dy) = dx + dy.
Question: Solve : (x+y) (dxdy) =dx+dy.[Hint: Substitutex+y=zafter separatingdxanddy] Solution: Given differential equation, (x+y) (dxdy) =dx+dy (x + y) dx (x y) dy = dx + dy (x + y) dy dy = dx (x + y) dx (x + y + 1) dy = (x + y 1) dx...
Read More →Find the general solution of
Question: Find the general solution ofy2dx+ (x2xy+y2)dy= 0. Solution: Given equation,y2dx+ (x2xy+y2)dy= 0...
Read More →Find the general solution of the differential equation
Question: Find the general solution of the differential equation Solution: Given differential equation, Thus, the above is the required solution of the given differential equation....
Read More →Prove the following
Question: Solve: $x^{2} \frac{d y}{d x}=x^{2}+x y+y^{2}$ Solution: Given equation,...
Read More →Find the equation of a curve passing through origin
Question: Find the equation of a curve passing through origin and satisfying the differential equation $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2}$ Solution: Given equation,...
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