Advertisements
Advertisements
प्रश्न
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Advertisements
उत्तर
`dy/ dx = (4x + y + 1)` ..(i)
Put 4x + y + 1 = t …(ii)
Differentiating w.r.t. x, we get
`4 + dy/dx = dt/ dx`
∴ `dy/dx = dt/ dx - 4` .... (iii)
Substituting (ii) and (iii) in (i), we get
`dt/ dx – 4 = t`
∴ `dt/ dx = t + 4`
∴ `dt/ (t + 4) = dx`
Integrating on both sides, we get
`intdt/(t+4) = int dx`
∴ log | t + 4 | = x + c
∴ log |(4x + y + 1) + 4 | = x + c
∴ log | 4x + y + 5 | = x + c …(iv)
When y = 1, x = 0, we have
log |4(0) + 1 + 5| = 0 + c
∴ c = log |6|
Substituting c = log |6| in (iv), we get
log |4x + y + 5| = x + log |6|
∴ log |4x + y + 5 | - log |6| = x
∴ log `|(4x+y+ 5) /6 | = x`,
which is the required particular solution.
APPEARS IN
संबंधित प्रश्न
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
Differential equation \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 3\] Function y = ex + e2x
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
(ey + 1) cos x dx + ey sin x dy = 0
(1 − x2) dy + xy dx = xy2 dx
(x + y) (dx − dy) = dx + dy
The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
The solution of the differential equation y1 y3 = y22 is
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
The solution of `dy/ dx` = 1 is ______.
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
Solve the differential equation:
dr = a r dθ − θ dr
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
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 value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
