Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[x + \left( \frac{dy}{dx} \right) = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
\[ \Rightarrow x + \left( \frac{dy}{dx} \right) = \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^\frac{1}{2} \]
Squaring both sides, we get
\[ \Rightarrow \left( x + \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \]
\[ \Rightarrow x^2 + 2x\frac{dy}{dx} + \left( \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \]
\[ \Rightarrow 2x\frac{dy}{dx} + x^2 = 1\]
In this differential equation, the order of the highest order derivative is 1 and the power is 1. So, it is a differential equation of order 1 and degree 1.
Hence, it is a linear differential equation.
Notes
The answer given in the book has some error. The solution here is created according to the question given in the book.
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
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 y = cx + 2c2 is a solution of the differential equation
Verify that \[y = e^{m \cos^{- 1} x}\] satisfies the differential equation \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - m^2 y = 0\]
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
Differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 2\] Function y = xex + ex
y (1 + ex) dy = (y + 1) ex dx
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
Solve the following differential equations:
\[\frac{dy}{dx} = \frac{y}{x}\left\{ \log y - \log x + 1 \right\}\]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
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.
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
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
For the following differential equation find the particular solution.
`(x + 1) dy/dx − 1 = 2e^(−y)`,
when y = 0, x = 1
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.
xdx + 2y dx = 0
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
Solve the following differential equation y2dx + (xy + x2) dy = 0
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
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
