Advertisements
Advertisements
प्रश्न
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Advertisements
उत्तर
y = log x + c
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = 1/x`
∴ `x ("d"y)/("d"x)` = 1
Again, differentiating w.r.t. x, we get
`x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Show that y = AeBx is a solution of the differential equation
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
x cos y dy = (xex log x + ex) dx
(1 − x2) dy + xy dx = xy2 dx
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
(y + xy) dx + (x − xy2) dy = 0
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
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.
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
Define a differential equation.
The differential equation satisfied by ax2 + by2 = 1 is
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Solve the following differential equation.
`dy/dx + y` = 3
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
x2y dx – (x3 + y3) dy = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 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
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
