Advertisements
Advertisements
Question
Advertisements
Solution
We have,
\[\left( x - 1 \right)\frac{dy}{dx} = 2 x^3 y\]
\[ \Rightarrow \frac{1}{y}dy = \frac{2 x^3}{x - 1}dx\]
Integrating both sides, we get
\[\int\frac{1}{y}dy = \int\frac{2 x^3}{x - 1}dx\]
\[ \Rightarrow \log \left| y \right| = 2\int\frac{x^3 - 1 + 1}{x - 1}dx\]
\[ \Rightarrow \log \left| y \right| = 2\int\frac{\left( x - 1 \right)\left( x^2 + x + 1 \right) + 1}{x - 1}dx\]
\[ \Rightarrow \log \left| y \right| = 2\int\left( x^2 + x + 1 \right)dx + 2\int\frac{1}{x - 1}dx\]
\[ \Rightarrow \log \left| y \right| = \frac{2}{3} x^3 + x^2 + 2x + \log \left| x - 1 \right| + C\]
\[\text{ Hence, }\log \left| y \right| = \frac{2}{3} x^3 + x^2 + 2x + \log \left| x - 1 \right| + \text{ C is the required solution }.\]
APPEARS IN
RELATED QUESTIONS
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`
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
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 = ex + 1
Differential equation \[\frac{dy}{dx} + y = 2, y \left( 0 \right) = 3\] Function y = e−x + 2
C' (x) = 2 + 0.15 x ; C(0) = 100
xy (y + 1) dy = (x2 + 1) dx
(1 − x2) dy + xy dx = xy2 dx
(y + xy) dx + (x − xy2) dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]
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.
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
\[\frac{dy}{dx} = \frac{y}{x} + \sin\left( \frac{y}{x} \right)\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
A bank pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
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.
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
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 = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
`dy/dx + 2xy = x`
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
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
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
