Solve the following differential equations: x dy dx =x + y - Vidyadip

Question

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

✓

Answer

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{x + vx}}{\text{x}}$
$\Rightarrow\ \text{v + x}\frac{\text{dv}}{\text{dx}}=1+\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1+\text{v}-\text{v}$
$\Rightarrow\ \text{x}\frac{\text{dv}}{\text{dx}}=1$
$\Rightarrow\ \text{dv}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\text{dv}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\ \text{v}=\log|\text{x}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \frac{\text{y}}{\text{x}}=\log|\text{x}|+\text{C}$
$\Rightarrow\ \text{y}=\text{x}\log|\text{x}|+\text{Cx}$
Hence, $\text{y}=\text{x}\log|\text{x}|+\text{Cx}$ is the required solution.

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

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Show that the differential equation of $(x-y) d y-(x+y) d x=0$ is homogeneous and solve it.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{e}^{3\text{x}}\sin4\text{x}\times2^\text{x}$

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 - 12x^2 + 18x + 15$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{1-\sqrt{\text{xa}}}\Big)$

Let $f : N \rightarrow N$ be a function as $f(x) = 9x^2 + 6x - 5.$ Show that $f : N \rightarrow S,$ where $S$ is the range of $f,$ is invertible. Find the inverse of $f$ and hence find $f^{-1}(43)$ and $f^{-1}(163).$

Evalute the following integrals:
$\int\frac{-\sin\text{x}+2\cos\text{x}}{2\sin\text{x}+\cos\text{x}}\text{dx}$

Find the particular solution of the differential equation $ \frac { d y } { d x } + y \cot x = 2 x + x ^ { 2 } \cot x $ ($ x \neq 0$) given that $y = 0,$ when $ x = \frac { \pi } { 2 }$.

A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?

Maximize Z = 18x + 10y
Subject to
$4\text{x}+\text{y}\geq20$
$2\text{x}+3\text{y}\geq30$
$\text{x},\text{y}\geq0$