Advertisements
Advertisements
प्रश्न
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Advertisements
उत्तर
`(x + 1) dy/dx -1 = 2e^(-y)`
∴ `(x + 1) dy /dx = 2/e^y + 1`
∴ `(x + 1) dy /dx = ((2+e^y))/e^y `
∴ `e^y /(2+e^y) dy= dx/(1+x)`
Integrating on both sides, we get
`int e^y/(2+e^y) dy = intdx/(1+x)`
∴ log| 2 + ey| = log |1 + x| + log |c|
∴ log |2 + ey| = log |c(1 + x)|
∴ 2 + ey = c (1 + x) ...(i)
When y = 0, x = 1, we have
2 + e0 = c (1 + 1)
∴ 2 + 1 = 2c
∴ c = `3/2`
Substituting c = `3/2` in (i), we get
`2 + e^y = 3/ 2 (1 + x)`
∴ 4 + 2ey = 3 + 3x
∴ 3x - 2ey - 1 = 0, which is the required particular solution.
APPEARS IN
संबंधित प्रश्न
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
C' (x) = 2 + 0.15 x ; C(0) = 100
y (1 + ex) dy = (y + 1) ex dx
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The solution of the differential equation y1 y3 = y22 is
Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).
The differential equation `y dy/dx + x = 0` represents family of ______.
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
The solution of `dy/ dx` = 1 is ______.
Solve the following differential equation y2dx + (xy + x2) dy = 0
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
