Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[\frac{dy}{dx} + 2x = e^{3x} \]
\[ \Rightarrow \frac{dy}{dx} = e^{3x} - 2x\]
\[ \Rightarrow dy = \left( e^{3x} - 2x \right)dx\]
Integrating both sides, we get
\[ \Rightarrow \int dy = \int\left( e^{3x} - 2x \right)dx\]
\[ \Rightarrow y = \frac{e^{3x}}{3} - 2\frac{x^2}{2} + C\]
\[ \Rightarrow y = \frac{e^{3x}}{3} - x^2 + C\]
\[ \Rightarrow y + x^2 = \frac{e^{3x}}{3} + C\]
\[So, y + x^2 = \frac{e^{3x}}{3} + \text{ C is defined for all }x \in R . \]
\[\text{ Hence,} y + x^2 = \frac{e^{3x}}{3} +\text{ C, where } x \in R,\text{ is the solution to the given differential equation }.\]
APPEARS IN
संबंधित प्रश्न
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
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
x cos y dy = (xex log x + ex) dx
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
y ex/y dx = (xex/y + y) dy
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
Find the equation to the curve satisfying x (x + 1) \[\frac{dy}{dx} - y\] = x (x + 1) and passing through (1, 0).
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 solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
Solve the differential equation:
`"x"("dy")/("dx")+"y"=3"x"^2-2`
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Choose the correct alternative:
Differential equation of the function c + 4yx = 0 is
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
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
