Advertisements
Advertisements
प्रश्न
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
Advertisements
उत्तर
We have,
\[y = e^{- x} + ax + b.............(1)\]
Differentiating both sides of equation (1) with respect to `x`, we have
\[\frac{dy}{dx} = - e^{- x} + a..............(2)\]
Differentiating both sides of equation (2) with respect to `x`, we have
\[\frac{d^2 y}{d x^2} = e^{- x} \]
\[ \Rightarrow e^x \frac{d^2 y}{d x^2} = 1\]
Hence, the given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
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
xy dy = (y − 1) (x + 1) dx
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
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).
x2 dy + y (x + y) dx = 0
2xy dx + (x2 + 2y2) dy = 0
(x + 2y) dx − (2x − y) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
In a culture, the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present?
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
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 \[\frac{dy}{dx} = \frac{y}{x} .\]
y2 dx + (x2 − xy + y2) dy = 0
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Form the differential equation from the relation x2 + 4y2 = 4b2
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the differential equation:
dr = a r dθ − θ dr
y2 dx + (xy + x2)dy = 0
Solve the differential equation `("d"y)/("d"x) + y` = e−x
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is
