Solve the following differential equation:x dy dx +y=xe^ x - Vidyadip

Question

Solve the following differential equation:$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\text{e}^{\text{x}}$

✓

Answer

Here, $\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}\text{e}^{\text{x}}$ $\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{e}^{\text{x}}$ It is a linear differential equation, comparing it with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ $\text{P}=\frac{1}{\text{x}}, \text{Q}=\text{e}^{\text{x}}$ I.F. $=\text{e}^{\int\text{Pdx}}$ $=\text{e}^{\int\frac{1}{\text{x}}\text{dx}}$ $=\text{e}^{\log\text{x}}$ $=\text{x}$Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$ $\text{y}\times(\text{x})=\int\text{e}^{\text{x}}\times\text{xdx + C}$ $\text{xy}=\text{x}\int\text{e}^{\text{x}}\text{dx}-\int(1\times\int\text{e}^{\text{x}}\text{dx})\text{dx + C}$ Using integration by parts $=\text{x}\text{e}^{\text{x}}-\int\text{e}^{\text{x}}\text{dx}+\text{C}$ $=\text{x}\text{e}^{\text{x}}-\text{e}^{\text{x}}+\text{C}$ $\text{xy}=(\text{x}-1)\text{e}^{\text{x}}+\text{C}$ $\text{y}=\Big(\frac{\text{x}-1}{\text{x}}\Big)\text{e}^{\text{x}}+\frac{\text{C}}{\text{x}},\text{x}>0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differential Equations→Differential Equations — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that $A^2 - 5A + 7I_2 = 0.$

A dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A, 12$ units of vitamin $B$ and $8$ units of vitamin $C.$ The vitamin contents of one $\ kg$ food is given below:

Food	Vitamin $A$	Vitamin $B$	Vitamin $C$
$X$	$1$	$2$	$3$
$Y$	$2$	$2$	$1$

One $\ kg$ of food $X$ costs $Rs. 16$ and one $\ kg$ of food $Y$ costs $Rs. 20.$ Find the least cost of the mixture which will produce the required diet?

If $x^m + y^n = 1,$ Prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$

A and B are two independent events. The probability that A and B occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Find the probability of occurrence of two events.

The vertices A, B, C of triangle ABC have respectively position vector $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ with respect to given origin O. Show that the point D where the bisector of $\angle{\text{A}}$ meets BC has position Vector $\vec{\text{d}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$, where $\beta=\big|\vec{\text{c}}-\vec{\text{a}}\big|$ and, $\gamma=\big|\vec{\text{a}}-\vec{\text{b}}\big|$.

Find the equation of the normal lines to the curve $3x^2 - y^2 = 8$ which are parallel to the line $x + 3y = 4.$

Let $\vec{\text{u}},\vec{\text{v}}$ and $\vec{\text{w}}$ be vectors such $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=\vec{0}.$ If $|\vec{\text{u}}|=3,|\vec{\text{v}}|=4$ and $|\vec{\text{w}}|=5,$ then find $\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}.$

If $A=\left[\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right]$ find $A^{-1}$, using $A^{-1}$ solve the system of equations
$2 x-3 y+5 z=11$
$3 x+2 y-4 z=-5$
$x+y-2 z=-3$

Without expanding, prove that:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\\\text{a}&\text{b}&\text{c}\end{vmatrix}=\begin{vmatrix}\text{y}&\text{b}&\text{q}\\\text{x}&\text{a}&\text{p}\\\text{z}&\text{c}&\text{r}\end{vmatrix}$

Prove that the curves $\text{y}^{2} = 4x \text{ and } x^{2} = 4y $ divided the area of square bounded by $x = 0, x = 4,y=4 \text{ and } y = 0$ into three equal parts.