Advertisements
Advertisements
Question
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Solution
We have,
\[y = \frac{1}{4} \left( x \pm a \right)^2 . . . . . \left( 1 \right)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = \frac{1}{4} \times 2\left( x \pm a \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2}\left( x \pm a \right)\]
Squaring both sides we get
\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = \left[ \frac{1}{2}\left( x \pm a \right) \right]^2 \]
\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = \frac{1}{4} \left( x \pm a \right)^2 \]
\[ \Rightarrow \left( \frac{dy}{dx} \right)^2 = y ............\left[\text{Using } \left( 1 \right) \right]\]
\[ \therefore y = \left( \frac{dy}{dx} \right)^2\]
Hence, the given function is the solution to 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 = AeBx is a solution of the differential equation
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
(1 + x2) dy = xy dx
xy (y + 1) dy = (x2 + 1) dx
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
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.
(y2 − 2xy) dx = (x2 − 2xy) dy
(x + 2y) dx − (2x − y) dy = 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:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
A population grows at the rate of 5% per year. How long does it take for the population to double?
The slope of the tangent at a point P (x, y) on a curve is \[\frac{- x}{y}\]. If the curve passes through the point (3, −4), find the equation of the curve.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
`dy/dx + y = e ^-x`
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve:
(x + y) dy = a2 dx
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
