Advertisements
Advertisements
Question
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Advertisements
Solution
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
RELATED QUESTIONS
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 y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
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
(ey + 1) cos x dx + ey sin x dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
(y + xy) dx + (x − xy2) dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + \left( 1 + y^2 \right) = 0\], given that y = 1, when x = 0.
(x2 − y2) dx − 2xy dy = 0
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
The solution of the differential equation y1 y3 = y22 is
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
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).
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
The function y = ex is solution ______ of differential equation
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
