Advertisements
Advertisements
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
Advertisements
Solution
`("d"y)/("d"x)` = x^{2}y + y
= (x^{2} + 1)y
∴ `1/y "d"y` = (x^{2} + 1) dx
Integrating on both sides, we get
`int 1/y "d"y = int(x^2 + 1) "d"x`
∴ log |y| = `x^3/3 + x + c`
APPEARS IN
RELATED QUESTIONS
Prove that :
`int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = pi/2, x != 0`
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 x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
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 = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
Show that y = e^{−x} + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
(e^{y} + 1) cos x dx + e^{y} sin x dy = 0
(1 + x) (1 + y^{2}) dx + (1 + y) (1 + x^{2}) dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y\cot x = 2\cos x, y\left( \frac{\pi}{2} \right) = 0\]
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 representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The solution of the differential equation y_{1} y_{3} = y_{2}^{2} is
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
If x^{m}y^{n} = (x + y)^{m+n}, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
Verify that the function y = e^{−3x} is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x^{2} – 3y^{2} – 4x = 8.
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).
The price of six different commodities for years 2009 and year 2011 are as follows:
Commodities | A | B | C | D | E | F |
Price in 2009 (₹) |
35 | 80 | 25 | 30 | 80 | x |
Price in 2011 (₹) | 50 | y | 45 | 70 | 120 | 105 |
The Index number for the year 2011 taking 2009 as the base year for the above data was calculated to be 125. Find the values of x andy if the total price in 2009 is ₹ 360.
Choose the correct option from the given alternatives:
The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Determine the order and degree of the following differential equations.
Solution | D.E. |
y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Form the differential equation from the relation x^{2 }+ 4y^{2 }= 4b^{2}
Solve the following differential equation.
`dy/dx + y = e ^-x`
Solve the following differential equation.
`dy/dx + y` = 3
Solve the following differential equation.
dr + (2r)dθ= 8dθ
Choose the correct alternative.
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
Choose the correct alternative.
The solution of `x dy/dx = y` log y is
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is e^{x}, then its solution is
Solve the differential equation:
dr = a r dθ − θ dr
Solve:
(x + y) dy = a^{2 }dx
Solve
`dy/dx + 2/ x y = x^2`
`dy/dx = log x`
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
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae^{5x} + Be^{–5x} is
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve the differential equation sec^{2}y tan x dy + sec^{2}x tan y dx = 0
Solve the differential equation `("d"y)/("d"x) + y` = e^{−x}
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the differential equation xdx + 2ydy = 0
Solve the differential equation (x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0
Solve the following differential equation `("d"y)/("d"x)` = x^{2}y + y
Solve: `("d"y)/("d"x) + 2/xy` = x^{2}
For the differential equation, find the particular solution (x – y^{2}x) dx – (y + x^{2}y) dy = 0 when x = 2, y = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x^{2} + 2y^{2}
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Solve the following differential equation y^{2}dx + (xy + x^{2}) dy = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x^{2} + xy − y^{2}
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
The function y = e^{x} is solution ______ of differential equation
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
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}
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)
Solution: `("d"y)/("d"x)` = cos(x + y) ......(1)
Put `square`
∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`
∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`
∴ (1) becomes `"dv"/("d"x) - 1` = cos v
∴ `"dv"/("d"x)` = 1 + cos v
∴ `square` dv = dx
Integrating, we get
`int 1/(1 + cos "v") "d"v = int "d"x`
∴ `int 1/(2cos^2 ("v"/2)) "dv" = int "d"x`
∴ `1/2 int square "dv" = int "d"x`
∴ `1/2* (tan("v"/2))/(1/2)` = x + c
∴ `square` = x + c
Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e^{2y} cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e^{2y} cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(-2y))/(-2)` = sin x + c_{1}
∴ e^{–2y} = – 2sin x – 2c_{1}
∴ `square` = c, where c = – 2c_{1 }
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
Solve the differential equation `"dy"/"dx"` = 1 + x + y^{2} + xy^{2}, when y = 0, x = 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
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
If `y = log_2 log_2(x)` then `(dy)/(dx)` =
A man is moving away from a tower 41.6 m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
The differential equation (1 + y^{2})x dx – (1 + x^{2})y dy = 0 represents a family of:
Solve the differential equation
`x + y dy/dx` = x^{2} + y^{2}
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.