For the differential equation, find the general solution:
\(\frac{dy}{dx} = \frac{1-\cos x}{1+\cos x}\)
\(y = 2\tan\frac{x}{2} - x + C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = \sqrt{4-y^2}\) \((-2<y<2)\)
\(y = 2\sin(x + C)\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} + y = 1\) \((y \ne 1)\)
\(y = 1 + Ae^{-x}\)
For the differential equation, find the general solution:
\(\sec^2 x\,\tan y\,dx + \sec^2 y\,\tan x\,dy = 0\)
\(\tan x\,\tan y = C\)
For the differential equation, find the general solution:
\((e^x + e^{-x})\,dy - (e^x - e^{-x})\,dx = 0\)
\(y = \log(e^x + e^{-x}) + C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = (1+x^2)(1+y^2)\)
\(\tan^{-1}y = x + \frac{x^3}{3} + C\)
For the differential equation, find the general solution:
\(y\log y\,dx - x\,dy = 0\)
\(y = e^{cx}\)
For the differential equation, find the general solution:
\(x^5\frac{dy}{dx} = -y^5\)
\(x^{-4} + y^{-4} = C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = \sin^{-1} x\)
\(y = x\sin^{-1}x + \sqrt{1-x^2} + C\)
For the differential equation, find the general solution:
\(e^x\tan y\,dx + (1-e^x)\sec^2 y\,dy = 0\)
\(\tan y = C(1 - e^x)\)
Find a particular solution satisfying the given condition:
\((x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x\), \(y = 1\) when \(x = 0\)
\(y = \frac{1}{4}\log\left[(x+1)^2(x^2+1)^3\right] - \frac{1}{2}\tan^{-1}x + 1\)
Find a particular solution satisfying the given condition:
\(x(x^2 - 1)\frac{dy}{dx} = 1\), \(y = 0\) when \(x = 2\)
\(y = \frac{1}{2}\log\left(\frac{x^2-1}{x^2}\right) - \frac{1}{2}\log\left(\frac{3}{4}\right)\)
Find a particular solution satisfying the given condition:
\(\cos\left(\frac{dy}{dx}\right) = a\) \((a \in \mathbb{R})\), \(y = 1\) when \(x = 0\)
\(\cos\left(\frac{y-2}{x}\right) = a\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx} = y\tan x\), \(y = 1\) when \(x = 0\)
\(y = \sec x\)
Find the equation of a curve passing through the point \((0,0)\) and whose differential equation is \(y' = e^x\sin x\).
\(2y - 1 = e^x(\sin x - \cos x)\)
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the solution curve passing through the point \((1,-1)\).
\(y - x + 2 = \log\left(x^2(y+2)^2\right)\)
Find the equation of a curve passing through the point \((0,-2)\) given that at any point \((x,y)\) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
\(y^2 - x^2 = 4\)
At any point \((x,y)\) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point \((-4,-3)\). Find the equation of the curve given that it passes through \((-2,1)\).
\((x+4)^2 = y + 3\)
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of the balloon after \(t\) seconds.
\((63t + 27)^{\frac{1}{3}}\)
In a bank, principal increases continuously at the rate of \(r\%\) per year. Find the value of \(r\) if Rs 100 doubles itself in 10 years \((\log_e 2 = 0.6931)\).
\(6.93\%\)
In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it be worth after 10 years \((e^{0.5} = 1.648)\).
Rs 1648
In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
\(\frac{2\log 2}{\log\left(\frac{11}{10}\right)}\)
The general solution of the differential equation \(\frac{dy}{dx} = e^{x+y}\) is
\(e^x + e^{-y} = C\)
\(e^x + e^y = C\)
\(e^{-x} + e^y = C\)
\(e^{-x} + e^{-y} = C\)