Form the differential equation of all circles
Question: Form the differential equation of all circles which pass through origin and whose centres lie on y-axis. Solution: The equation of circles which pass through the origin and whose centre lies on the y-axis is given by, (x 0)2+ (y a)2= a2, where (0, a) is the centre x2+ y2+ a2 2ay = a2 $\left(x^{2}-y^{2}\right) \frac{d y}{d x}-2 x y=0$...
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Question: Form the differential equation having y = (sin1x)2+ A cos1x + B, where A and B are arbitrary constants, as its general solution. Solution:...
Read More →If y(t) is a solution of and y (0) = – 1,
Question: Ify(t) is a solution of andy(0) = 1, then show thaty(1) = -1/2. Solution:...
Read More →If y(x) is a solution of and y (0) = 1,
Question: Ify(x) is a solution of and y (0) = 1, then find the value of y(/2). Solution:...
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Question: Find the general solution of (x+ 2y3) dy/dx = y. Solution:...
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Question: Solve the differential equation dy/dx = 1 +x+y2+xy2, wheny= 0,x= 0. Solution: Given equation, dy/dx = 1 +x+y2+xy2...
Read More →Solve: ydx – xdy = x2ydx.
Question: Solve:ydxxdy=x2ydx. Solution: Given equation, ydx xdy = x2ydx y dx x2y dx = xdy y (1 x2) dx = xdy...
Read More →Solve the differential equation
Question: Solve the differential equation $\frac{d y}{d x}+1=e^{x+y}$ Solution: Given differential equation, dy/dx + 1 = ex+y Substituting x + y = t and differentiating w.r.t. x, we have...
Read More →Find the general solution of
Question: Find the general solution of $\frac{d y}{d x}+a y=e^{m x}$ Solution: Given equation, dy/dx + ay = emx Solving for linear differential equation of first order, we have P = a and Q = emx...
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Question: Solve the differential equation $\frac{d y}{d x}+2 x y=y$ Solution:...
Read More →Solve the differential equation
Question: Solve the differential equation $\left(x^{2}-1\right) \frac{d y}{d x}+2 x y=\frac{1}{x^{2}-1}$ Solution:...
Read More →Given that and y = 0 when x = 5.
Question: Given that andy= 0 whenx= 5. Find the value ofxwheny= 3. $\frac{d y}{d x}=e^{-2 y}$ Solution:...
Read More →Find the differential equation of all non-vertical lines in a plane.
Question: Find the differential equation of all non-vertical lines in a plane. Solution: We know that, the equation of all non-vertical lines are y = mx + c On differentiating w.r.t. x, we get dy/dx = m Again, on differentiating w.r.t. x, we have d2y/dx2= 0 Thus, the required equation is d2y/dx2= 0....
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Question: Find the solution of $\frac{d y}{d x}=2^{y-x}$ Solution:...
Read More →Prove the following identities.
Question: $\int_{0}^{\frac{\pi}{2}} \frac{\tan x d x}{1+m^{2} \tan ^{2} x}$ Solution:...
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Question: $\int_{0}^{2}\left(x^{2}+3\right) d x$ Solution:...
Read More →Prove the following identities.
Question: $\int \frac{d x}{x \sqrt{x^{4}-1}}$ $\left(\right.$ Hint : Put $\left.x^{2}=\sec \theta\right)$ Solution:...
Read More →Prove the following identities.
Question: $\int \frac{\cos x-\cos 2 x}{1-\cos x} d x$ Solution:...
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Question: $\int \frac{\sin ^{6} x+\cos ^{6} x}{\sin ^{2} x \cos ^{2} x} d x$ Solution:...
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Question: $\int \frac{(\cos 5 x+\cos 4 x)}{1-2 \cos 3 x} d x$ Solution:...
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Question: $\int \frac{\sin ^{-1} x}{\left(1-x^{2}\right)^{\frac{3}{2}}} d x$ Solution:...
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