Advertisements
Advertisements
प्रश्न
Show that y = C x + 2C2 is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]
Advertisements
उत्तर
We have,
\[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0 . . . . . \left( 1 \right)\]
Now,
y = C x + 2C2
\[\Rightarrow\frac{dy}{dx}=C\]
\[\text{Putting }\frac{dy}{dx} = C\text{ and }y = Cx + 2 C^2\text{ in (1), we get}\]
\[\text{LHS }= 2 \left( C \right)^2 + x\left( C \right) - \left( Cx + 2 C^2 \right)\]
\[ = 2 C^2 + xC - xC - 2 C^2 \]
\[ = 0 =\text{ RHS}\]
Thus, y = C x + 2C2 is the solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Write the integrating factor of the following differential equation:
(1+y2) dx−(tan−1 y−x) dy=0
Find the differential equation of the family of lines passing through the origin.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
`x/a + y/b = 1`
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y2 = a (b2 – x2)
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e– 2x
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`
The general solution of the differential equation `(y dx - x dy)/y = 0` is ______.
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is ______.
Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where a and b are arbitrary constants.
Form the differential equation having \[y = \left( \sin^{- 1} x \right)^2 + A \cos^{- 1} x + B\], where A and B are arbitrary constants, as its general solution.
Verify that xy = a ex + b e−x + x2 is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]
Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]
Find the differential equation corresponding to y = ae2x + be−3x + cex where a, b, c are arbitrary constants.
From x2 + y2 + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.
\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]
\[\frac{dy}{dx} + 4x = e^x\]
\[\frac{dy}{dx} = x^2 e^x\]
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]
\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]
(1 + x) y dx + (1 + y) x dy = 0
x cos2 y dx = y cos2 x dy
cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy
(1 − x2) dy + xy dx = xy2 dx
A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.
Find the general solution of the following differential equation:
`x (dy)/(dx) = y - xsin(y/x)`
The general solution of the differential equation `(dy)/(dx) + x/y` = 0 is
General solution of tan 5θ = cot 2θ is
Solution of the equation 3 tan(θ – 15) = tan(θ + 15) is
The number of arbitrary constant in the general solution of a differential equation of fourth order are
Which of the following equations has `y = c_1e^x + c_2e^-x` as the general solution?
Find the general solution of differential equation `(dy)/(dx) = (1 - cosx)/(1 + cosx)`
What is the general solution of differential equation `(dy)/(dx) = sqrt(4 - y^2) (-2 < y < 2)`
The general solution of the differential equation y dx – x dy = 0 is ______.
Solve the differential equation: y dx + (x – y2)dy = 0
The general solution of the differential equation ydx – xdy = 0; (Given x, y > 0), is of the form
(Where 'c' is an arbitrary positive constant of integration)
