Solve the following differential equation:x dy dx =y - x ( y ax )… - Vidyadip

Question

Solve the following differential equation:$\text{x }\frac{\text{dy}}{\text{dx}}=\text{y - x}\tan\Bigg(\frac{\text{y}}{\text{ax}}\Bigg).$

✓

Answer

$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\tan\Bigg(\frac{\text{y}}{\text{x}}\Bigg).........\text{(i)}$ Let y = vx $\Rightarrow$ $\frac{\text{dy}}{\text{dx}}=\text{v + x }\frac{\text{dv}}{\text{dx}}$ $\therefore\text{(i) becomes v + x }\frac{\text{dv}}{\text{dx}}=\text{v - tan v}$$\Rightarrow-\cot\text{v dv}=\frac{\text{dx}}{\text{x}}$
log | cosec v | = log | cx |
$\Rightarrow\text{ c x }=\text{cosec }\Bigg(\frac{\text{y}}{\text{x}}\Bigg)$
$\text{OR }\Bigg(\text{x sin}\Bigg(\frac{\text{y}}{\text{x}}\Bigg)=\text{c}\Bigg).$

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

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.

Prove the following results:
$\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}=\tan^{-1}\frac{63}{16}$

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of $₹\ 6,000$. Three times the award money for Hard work added to that given for honesty amounts to $₹\ 11,000$. The award money given for Honesty and Hardwork together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Three vectors $\vec{a}, \vec{b}$ and $\vec c$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. Evaluate the quantity $\mu=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$, if $|\vec{a}|=3,|\vec{b}|=4$ and $|\vec{c}|=2$.

Is the function f defined by
$\text{f(x)} = \begin{cases}\text{x}, \text{if}\ \text{x}\leq1\\5, \text{if}\ \text{x} > 1\end{cases}$
continuous at x = 0? At x = 1? At x = 2?

Evaluate the following integrals:
$\int(3\text{x}+1)\sqrt{4-3\text{x}-2\text{x}^2}\text{dx}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=\text{x}^2\cos^2\text{x}$

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio $\text{1 : 2 : 4. }$The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.