Solve the following initial value problems: (xy-y^2)dx-x^2dy=0,y(… - Vidyadip

Question

Solve the following initial value problems:
$(\text{xy}-\text{y}^2)\text{dx}-\text{x}^2\text{dy}=0,\text{y}(1)=1$

✓

Answer

$(\text{xy}-\text{y}^2)\text{dx}-\text{x}^2\text{dy}=0,\text{y}(1)=1$
It is a homogeneous equation
Put y = vx
and $\frac{\text{dy}}{\text{dx}}=\text{v + x}\frac{\text{dv}}{\text{dx}}$
So,
$\text{v + x}\frac{\text{dv}}{\text{dx}}=\frac{\text{xvx}-\text{v}^2\text{x}^2}{\text{x}^2}$
$\text{x}\frac{\text{dv}}{\text{dx}}=\text{v}-\text{v}^2-\text{v}$
$\text{x}\frac{\text{dv}}{\text{dx}}=-\text{v}^2$
$-\int\frac{1}{\text{v}^2}\text{dv}=\int\frac{\text{dx}}{\text{x}}$
$-\Big(-\frac{1}{\text{v}}\Big)=\log|\text{x}|+\text{C}$
$\frac{\text{x}}{\text{y}}=\log|\text{x}|+\text{C}\ \dots(\text{i})$
Put y = 1, x = 1
1 = C
Using equation (1),
$\text{x}=\text{y}\big[\log|\text{x}|+1\big]$
$\text{y}=\frac{\text{x}}{\big[\log|\text{x}|+1\big]}$

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 "4 Marks" 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

Let f: N → N be defined by ${f(n)}=\begin{cases}\frac{\text{n}+1}{2},\text{ if n is odd}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for all}\text{ n }\in\text{ N}.\\\frac{\text{n}}{2}\ \ \ \ ,\text{if n is even}\end{cases}$ State whether the function f is bijective. Justify your answer.

A vector $\vec{\text{r}}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{\text{r}}$ is $2\sqrt{3}$ units, find $\vec{\text{r}}.$

Find the area of the region bounded by the paraola y² = 4ax and line x = a.

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\sin\text{x}}{\text{e}^{\text{x}}}\text{ on }0\leq\text{x}\leq\pi$

Prove the following results:
$2\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{8}=\tan^{-1}\frac{4}{7}$

Evaluate the following intregals:
$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+2\text{x}-1}}\ \text{dx}$

Prove that:
$\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\frac{\pi}{2}\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}$

Find the value of $\theta$ satisfying $\begin{bmatrix}1&1&\sin3\theta\\-4&3&\cos2\theta\\7&-7&-2\end{bmatrix}=0.$

$\int\frac{1}{\text{x}^{\frac{1}{3}}\big(\text{x}^{\frac{1}{3}}-1\big)}\text{dx}$

Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)

$\frac{\text{dy}}{\text{dx}}+3\text{y}=\text{e}^{\text{mx}},$ m is given real number.