Advertisements
Advertisements
प्रश्न
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Advertisements
उत्तर
We have,
y'' − y' = 0 ............(1)
Now,
y = ex +1
⇒ y'= ex
⇒ y'' = ex
Putting the above values in (1), we get
LHS = ex − ex = 0 = RHS
Thus, y = ex + 1 is the solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
xy (y + 1) dy = (x2 + 1) dx
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 differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
(x2 − y2) dx − 2xy dy = 0
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
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).
Solve the following differential equation.
`dy/dx = x^2 y + y`
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
`(x + y) dy/dx = 1`
The solution of `dy/ dx` = 1 is ______.
`xy dy/dx = x^2 + 2y^2`
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
Which of the following is an example of an ordinary differential equation?
