Advertisements
Advertisements
प्रश्न
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Advertisements
उत्तर
We have,
\[y = a x^3 + b x^2 + c ...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = 3a x^2 + 2bx ...........(2)\]
Differentiating both sides of (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = 6ax + 2b..............(3)\]
Differentiating both sides of (3) with respect to x, we get
\[\frac{d^3 y}{d x^3} = 6a\]
Hence, the given function is the solution to the given differential equation.
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
(ey + 1) cos x dx + ey sin x dy = 0
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
x2 dy + y (x + y) dx = 0
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
Find the equation of the curve which passes through the point (3, −4) and has the slope \[\frac{2y}{x}\] at any point (x, y) on it.
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.
The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact. Find the particular curve through the point (1, 1).
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
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).
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
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
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
The solution of `dy/ dx` = 1 is ______.
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve the differential equation:
`e^(dy/dx) = x`
y2 dx + (xy + x2)dy = 0
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
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
