Solve the following differential equation: cosec x y dy dx +x^2y^… - Vidyadip

Question

Solve the following differential equation:
$\text{cosec }x\ \log\text{ y}\frac{\text{dy}}{\text{d}x}+x^2\text{y}^2=0$

✓

Answer

$\text{cosec }x.\ \log\text{y}\frac{\text{dy}}{\text{d}x}=-x^2\text{y}^2$$\Rightarrow\frac{\log\text{ y}}{\text{y}^2}\text{ dy}=-\text{x}^2\sin\text{x dx}$
Integrating both sideswe get
$\Rightarrow-\frac{\log\text{ y}}{\text{y}}\frac{1}{\text{y}}=-[-\text{x}^2\cos\text{x}+2\int\text{x}\cos\text{x dx}]$
$=-[-\text{x}^2\cos\text{x}+2(\text{x}\sin\text{x}-\int1.\sin\text{x dx}]$
$\therefore\frac{\log\text{y}}{\text{y}}-\frac{1}{\text{y}}=-\text{x}^2\cos\text{x}+2\text{x}\sin\text{x}+2\cos\text{x}+\text{c}$

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

Differentiate the following functions with respect to x:
$(\tan\text{x})^\frac{1}{\text{x}}$

Find all points of discontinuity of f, where f is defined by:
$\text{f(x)}= \begin{cases}\text{x} + 1,\ \ \text{if x}\geq 1 \\\text{x}^2 + 1,\ \text{if x}<1\end{cases}$

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\sin\text{x}\cos^4\text{x}\text{ dx}$

Solve the differential equation: $(\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{e}^{-\text{y}}-1;\ \text{y(0)}=0.$

If $\text{x}-\text{e}^{\frac{\text{x}}{\text{y}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}-\text{y}}{\text{x}\log\text{x}}$

Discuss the continuity of $\text{f(x)}=\sin|\text{x}|$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=1-\text{x + y}-\text{xy}$

A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = The card drawn is a king or queen,
B = the card drawn is a queen or jack.

Find the area of the region bounded by the curve $y^2 = 4x$ and the line $x = 3.$

Find the direction cosines of the line $\frac{\text{x}+2}{2}=\frac{2\text{y}-7}{6}=\frac{5-\text{z}}{6}.$ Also, find the vector equation of the line through the point A(-1, 2, 3) and parallel to the given line.