Advertisements
Advertisements
Question
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Options
`(d^2y)/dx^2 + 2 dy/dx = 0`
`x(d^2y)/dx^2 + 2 dy/dx = 0`
`(d^2y)/dx^2 -2 dy/dx = 0`
`x(d^2y)/dx^2 -2 dy/dx = 0`
Advertisements
Solution
The differential equation of `y = k_1 + k_2/x` is `x(d^2y)/dx^2 + 2 dy/dx = 0`
Explanation
`y = k_1 + k_2/x`
∴ xy = xk1 + k2
Differentiating w.r.t. x, we get
`y+x dy/dx = k_1`
Again, differentiating w.r.t. x, we get
`dy/dx + dy/dx + x (d^2y)/dx^2 = 0`
∴ `x (d^2y)/dx^2 + 2 dy/dx = 0`
APPEARS IN
RELATED QUESTIONS
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
C' (x) = 2 + 0.15 x ; C(0) = 100
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Solve the following differential equation : \[\left( \sqrt{1 + x^2 + y^2 + x^2 y^2} \right) dx + xy \ dy = 0\].
y2 dx + (x2 − xy + y2) dy = 0
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
`(dθ)/dt = − k (θ − θ_0)`
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
