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

Prove the following identities:
$\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}=4\text{xyz}$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}|\text{x}-\text{a|}\sin(\frac{1}{\text{x}-\text{a}}), &\text{for} \text{ x} \neq\text{a}\\0,&\text{for} \text{ x} = \text{a}\end{cases}\text{ at x}=0$

Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^4+1)}\ \text{dx}$

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sin\Big(\frac{\text{y}}{\text{x}}\Big)$

Differentiate the following functions from first principles:
$\text{e}^{\cos\text{x}}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^2=4\text{y}\text{ at }(2,1)$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x + y}}$

Evaluate the following integrals:
$\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$