Advertisements
Advertisements
प्रश्न
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Advertisements
उत्तर
y = P eax + Q ebx
Differentiating w.r.t x, we get:
`dy/dx=Pae^(ax)+Qbe^(bx)..................(1)`
`(a+b)dy/dx=(a+b)(Pae^(ax)+Qbe^(bx))`
`(a+b)dy/dx=Pa^2e^(ax)+Qb^2e^(bx)+ab(Pe^(ax)+Qe^(bx))`
`(a+b)dy/dx=Pa^2e^(ax)+Qb^2e^(bx)+aby`
`-[-(a+b)dy/dx+aby]=Pa^2e^(ax)+Qb^2e^(bx)........(2)`
Differentiating (1) w.r.t. x, we get:
`(d^y)/(dx^2)=Pa^2e^(ax)+Qb^2e^(bx)................(3)`
Subtracting (2) from (3), we get:
`(d^y)/(dx^2)-(a+b)dy/dx+aby=Pa^2e^(ax)+Qb^2e^(bx)-Pa^2e^(ax)-Qb^2e^(bx)`
`(d^y)/(dx^2)-(a+b)dy/dx+aby=0`
Hence proved.
APPEARS IN
संबंधित प्रश्न
If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`
Find the particular solution of differential equation:
`dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0`
Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy' = `y + x sqrt (x^2 - y^2)` (x ≠ 0 and x > y or x < -y)
The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is
Find the general solution of the differential equation \[x \cos \left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x .\]
Find the particular solution of the differential equation `(1+y^2)+(x-e^(tan-1 )y)dy/dx=` given that y = 0 when x = 1.
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\] given that
\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]
(x + y − 1) dy = (x + y) dx
\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]
\[\left( 1 + y^2 \right) + \left( x - e^{- \tan^{- 1} y} \right)\frac{dy}{dx} = 0\]
`2 cos x(dy)/(dx)+4y sin x = sin 2x," given that "y = 0" when "x = pi/3.`
For the following differential equation, find the general solution:- `y log y dx − x dy = 0`
Solve the following differential equation:-
\[\left( x + 3 y^2 \right)\frac{dy}{dx} = y\]
Find a particular solution of the following differential equation:- \[\left( 1 + x^2 \right)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}; y = 0,\text{ when }x = 1\]
The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
Solution of differential equation xdy – ydx = 0 represents : ______.
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.
The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.
The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.
Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.
Which of the following differential equations has `y = x` as one of its particular solution?
Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.
