Advertisements
Advertisements
प्रश्न
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Advertisements
उत्तर
ax 2 + by 2 = 5
Differentiating w.r.t. x, we get
`2ax +2by dy/dx = 0` ....(i)
Again, differentiating w.r.t. x, we get
`2a + 2b(dy/dx)^2 + 2by ((d^2y)/dx^2) = 0` ........(ii)
From (i), we get
`a = - (by)/x(dy/dx)`
Substituting the value of a in (ii), we get
`- 2(by)/x(dy/dx) + 2b(dy/dx)^2 + 2by((d^2y)/dx^2) = 0`
∴`- y/x(dy/dx) + (dy/dx)^2 + y((d^2y)/dx^2) = 0`
∴`- y(dy/dx) + x(dy/dx)^2 + xy((d^2y)/dx^2) = 0`
∴`x y((d^2y)/dx^2) + x(dy/dx)^2 =y((dy)/dx) `
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Show that y = e−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
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.
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
The integrating factor of the differential equation `dy/dx - y = x` is e−x.
y2 dx + (xy + x2)dy = 0
x2y dx – (x3 + y3) dy = 0
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation `("d"y)/("d"x) + y` = e−x
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
