Advertisements
Advertisements
Question
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
Advertisements
Solution
Given equation is `"dy"/"dx"` = 1 + x + y2 + xy2
⇒ `"dy"/"dx"` = 1(1 + x) + y2(1 + x)
⇒ `"dy"/"dx"` = (1 + x)(1 + y2)
⇒ `"dy"/(1 + y^2)` = (1 + x)dx
Integrating both sides, we get
`int "dy"/(1 + y^2) = int(1 + x)"d"x`
⇒ `tan^-1y = x + x^2/2 + "c"`
Put x = 0 and y = 0
We get tan–1(0) = 0 + 0 + c
⇒ c = 0
∴ tan–1y = `x + x^2/2`
⇒ y = `tan(x + x^2/2)`
Hence, the required solution is y = `tan(x + x^2/2)`.
APPEARS IN
RELATED QUESTIONS
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
xy (y + 1) dy = (x2 + 1) dx
x cos2 y dx = y cos2 x dy
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
Solve the following differential equation:
\[\left( 1 + y^2 \right) \tan^{- 1} xdx + 2y\left( 1 + x^2 \right)dy = 0\]
Solve the differential equation \[x\frac{dy}{dx} + \cot y = 0\] given that \[y = \frac{\pi}{4}\], when \[x=\sqrt{2}\]
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.
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
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.
2xy dx + (x2 + 2y2) dy = 0
Solve the following initial value problem:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]
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.
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
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 curves y = a sin (x + b), where a, b are arbitrary constant.
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
The differential equation of `y = k_1e^x+ k_2 e^-x` 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.
`xy dy/dx = x^2 + 2y^2`
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
