Solve: dydx+2xy = x2 - Mathematics and Statistics

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.

Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.

Verify that y² = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]

Verify that \[y = ce^{tan^{- 1}} x\] is a solution of the differential equation \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + \left( 2x - 1 \right)\frac{dy}{dx} = 0\]

Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]

Function y = e^x + 1

\[\left( x + 2 \right)\frac{dy}{dx} = x^2 + 3x + 7\]

\[\frac{dy}{dx} = \cos^3 x \sin^2 x + x\sqrt{2x + 1}\]

(sin x + cos x) dy + (cos x − sin x) dx = 0

\[\sin^4 x\frac{dy}{dx} = \cos x\]

\[\sqrt{1 - x^4} dy = x\ dx\]

\[\sqrt{a + x} dy + x\ dx = 0\]

\[\frac{dy}{dx} = \frac{1 + y^2}{y^3}\]

\[\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\]

\[\frac{dy}{dx} = \left( e^x + 1 \right) y\]

\[\left( x - 1 \right)\frac{dy}{dx} = 2 x^3 y\]

xy (y + 1) dy = (x² + 1) dx

x cos y dy = (xe^x log x + e^x) dx

\[\frac{dy}{dx} = \frac{x e^x \log x + e^x}{x \cos y}\]

\[\frac{dy}{dx} = \frac{e^x \left( \sin^2 x + \sin 2x \right)}{y\left( 2 \log y + 1 \right)}\]

\[\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}\]

\[\cos x \cos y\frac{dy}{dx} = - \sin x \sin y\]

(y + xy) dx + (x − xy²) dy = 0

\[\frac{dy}{dx} = 1 - x + y - xy\]

(y² + 1) dx − (x² + 1) dy = 0

\[\frac{dy}{dx} = e^{x + y} + e^{- x + y}\]

Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]

Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]

Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]

\[\frac{dy}{dx} = y \tan 2x, y\left( 0 \right) = 2\]

\[2x\frac{dy}{dx} = 3y, y\left( 1 \right) = 2\]

\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]

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 (e^0.5 = 1.648).

In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.

\[\frac{dy}{dx} = \left( x + y \right)^2\]

\[\frac{dy}{dx} = \tan\left( x + y \right)\]

\[\left( x + y + 1 \right)\frac{dy}{dx} = 1\]

(x² − y²) dx − 2xy dy = 0

\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]

2xy dx + (x² + 2y²) dy = 0

\[\frac{dy}{dx} = \frac{x}{2y + x}\]

Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]

\[\left[ x\sqrt{x^2 + y^2} - y^2 \right] dx + xy\ dy = 0\]

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:-

\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]

Solve the following initial value problem:-

\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]

Solve the following initial value problem:-

\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?

If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.

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.

Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan⁻¹\[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.

Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e^2x.

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.

Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is \[y^2 - 2xy\frac{dy}{dx} - x^2 = 0\], and hence find the curve.

The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.

Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?

Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.

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).

If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.

Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?

What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?

Which of the following differential equations has y = C₁ e^x + C₂ e⁻^x as the general solution?

Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0

Find the differential equation whose general solution is

x³ + y³ = 35ax.

Form the differential equation from the relation x²+ 4y²= 4b²

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.

(x² − y²) dx + 2xy dy = 0

Solve the following differential equation.

`dy/dx + 2xy = x`

Solve the following differential equation.

dr + (2r)dθ= 8dθ

Choose the correct alternative.

The differential equation of y = `k_1 + k_2/x` is

Choose the correct alternative.

The solution of `x dy/dx = y` log y is

State whether the following is True or False:

The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.

Solve the differential equation:

dr = a r dθ − θ dr

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

Solve the following differential equation `("d"y)/("d"x)` = x²y + y

For the differential equation, find the particular solution

`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0

A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution

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

The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`

Given that `"dy"/"dx"` = ye^x and x = 0, y = e. Find the value of y when x = 1.

Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is e^x.

Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]

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)` =

Solve the differential equation

`y (dy)/(dx) + x` = 0

Solve the differential equation

`x + y dy/dx` = x² + y²

Solve: dydx+2xy = x2 - Mathematics and Statistics

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Solve: dydx+2xy = x2 - Mathematics and Statistics

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