Advertisements
Advertisements
प्रश्न
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Advertisements
उत्तर
We have,
\[y = 2\left( x^2 - 1 \right) + c e^{- x^2}...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = 4x - c e^{- x^2} 2x\]
\[ = 2x\left[ 2 - c e^{- x^2} \right]\]
\[ = - 2x\left[ 2 x^2 - 2 + c e^{- x^2} - 2 x^2 \right]\]
\[ = - 2x\left[ 2\left( x^2 - 1 \right) + c e^{- x^2} - 2 x^2 \right]\]
\[ = - 2x\left[ y - 2 x^2 \right] .............\left[\text{Using }\left( 1 \right) \right]\]
\[ \Rightarrow \frac{dy}{dx} = - 2xy + 4 x^3 \]
\[ \Rightarrow \frac{dy}{dx} + 2xy = 4 x^3\]
Hence, the given function is the solution to the given differential equation.
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
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 \[\frac{d^2 y}{d x^2} + y = 0\]
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
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
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
y (1 + ex) dy = (y + 1) ex dx
(y2 + 1) dx − (x2 + 1) dy = 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.
x2 dy + y (x + y) dx = 0
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
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 of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The solution of the differential equation \[\frac{dy}{dx} - \frac{y\left( x + 1 \right)}{x} = 0\] is given by
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-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 particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
