Advertisements
Advertisements
Question
Advertisements
Solution
We have,
\[\frac{dy}{dx} = e^{x + y} + e^y x^3 \]
\[ \Rightarrow \frac{dy}{dx} = e^x e^y + e^y x^3 \]
\[ \Rightarrow \frac{dy}{dx} = e^y \left( e^x + x^3 \right)\]
\[ \Rightarrow \left( e^x + x^3 \right) dx = \frac{1}{e^y}dy\]
Integrating both sides, we get
\[\int\left( e^x + x^3 \right) dx = \int\frac{1}{e^y}dy\]
\[ \Rightarrow e^x + \frac{x^4}{4} = - e^{- y} + C\]
\[ \Rightarrow e^x + e^{- y} + \frac{x^4}{4} = C\]
\[\text{ Hence, } e^x + e^{- y} + \frac{x^4}{4} =\text{ C is the required solution .} \]
APPEARS IN
RELATED QUESTIONS
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
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
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
y ex/y dx = (xex/y + y) dy
3x2 dy = (3xy + y2) dx
Solve the following initial value problem:-
\[\frac{dy}{dx} - 3y \cot x = \sin 2x; y = 2\text{ when }x = \frac{\pi}{2}\]
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
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.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
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 of the family of circles having centre on y-axis and radius 3 unit.
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
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
`dy/dx + 2/ x y = x^2`
y2 dx + (xy + x2)dy = 0
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
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
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constant is called ______ solution
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Solve the differential equation `"dy"/"dx" + 2xy` = y
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
