Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} = e^{x + y} + e^{- x + y} \]
\[ \Rightarrow \frac{dy}{dx} = e^y \left( e^x + e^{- x} \right)\]
\[ \Rightarrow e^{- y} dy = \left( e^x + e^{- x} \right) dx\]
Integrating both sides, we get
\[\int e^{- y} dy = \int\left( e^x + e^{- x} \right) dx\]
\[ \Rightarrow - e^{- y} = e^x - e^{- x} + C\]
\[ \Rightarrow e^{- x} - e^{- y} = e^x + C\]
\[\text{ Hence, } e^{- x} - e^{- y} = e^x + C\text{ is the required solution.} \]
APPEARS IN
संबंधित प्रश्न
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
x cos y dy = (xex log x + ex) dx
x cos2 y dx = y cos2 x dy
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
The differential equation `y dy/dx + x = 0` represents family of ______.
Find the differential equation whose general solution is
x3 + y3 = 35ax.
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
The solution of `dy/dx + x^2/y^2 = 0` is ______
Solve: `("d"y)/("d"x) + 2/xy` = x2
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
