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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the shortest distance between the following pairs of lines whose cartesian equation are:
$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\text{z}$ and $\frac{\text{x}+2}{3}=\frac{\text{y}-2}{1};\text{z}=2$

Show that the points (3, 4), (-5, 16) and (5, 1) are collinear.

A discrete random variable $X$ has the probability distribution given as below:

$X$	$0.5$	$1$	$1.5$	$2$
$P(X)$	$k$	$k^2$	$2k^2$	$k$

Find the value of $k.$
Determine the mean of the distribution.

If AD is the median of $\triangle\text{ABC},$ using vectors, prove that $\text{AB}^2+\text{AC}^2=2\big(\text{AD}^2+\text{CD}^2\big).$

Evaluate the following integrals:
$\int\text{x}\sin^3\text{x dx}$

Using the method of interation, find the area of the region bounded by the following lines:
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.

If $\text{A}=\begin{bmatrix} 3\\5\\2\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0&4\end{bmatrix},$ verify that $(AB)^T = B^TA^T$.

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\text{x}+\frac{\text{a}^{2}}{\text{x}}$

Find: $\int(\text{3x + 5)}\sqrt{5+4x-2x^2}\text{dx}$

A line passes through (2, –1, 3) and is perpendicular to the lines$\overrightarrow{\text{r}}= (\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}) + \lambda(2 \hat{\text{i}} - 2\hat{\text{j}} + \hat{\text{k}})\text{ and }\overrightarrow{\text{r}} = (2\hat{\text{i}} -\hat{\text{j}} - 3\hat{\text{k}}) + \mu(\hat{\text{i}} + 2 \hat{\text{j}} + 2\hat{\text{k}}).$Obtain its equation in vector and cartesian form.