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

Question

Solve the following differential equations:

$\text{cosec x}\log\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}^2\text{y}^2=0$

✓

Answer

$\text{cosec x}\log\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}^2\text{y}^2=0$
$\Rightarrow\text{cosec x }\log\text{y}\frac{\text{dy}}{\text{dx}}=-\text{x}^2\text{y}^2$
$\Rightarrow\frac{1}{\text{y}^2}\log\text{ y dy}=-\frac{\text{x}^2}{\text{cosec x}}\text{dx}$
$\Rightarrow\frac{1}{\text{y}^2}\log\text{ y dy}=-\text{x}^2\sin\text{x dx}$
$\Rightarrow\int\frac{1}{\text{y}^2}\log\text{ y dy}=-\int\text{x}^2\sin\text{x dx}$
$\Rightarrow-\frac{\log\text{y}}{\text{y}}+\int\frac{1}{\text{y}}\times\frac{1}{\text{y}}=-\Big[-\text{x}^2\cos\text{x}+\int2\text{x}\cos\text{x dx}\Big]+\text{C}$
$\Rightarrow-\frac{\log\text{y}}{\text{y}}-\frac{1}{\text{y}}=-\Big[-\text{x}^2\cos\text{x}+2\text{x}\sin\text{x}-2\int\sin\text{x dx}\Big]+\text{C}$
$\Rightarrow-\Big(\frac{1+\log\text{y}}{\text{y}}\Big)=-\big[-\text{x}^2\cos\text{x}+2\text{x}\sin\text{x}+2\cos\text{x dx}\big]+\text{C}$
$\Rightarrow-\Big(\frac{1+\log\text{y}}{\text{y}}\Big)-\text{x}^2\cos\text{x}+2(\text{x}\sin\text{x}+\cos\text{x})=\text{C}$

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 "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

Evaluate the following integrals:
$\int\frac{\text{x}^2}{(\text{x}^2+\text{a}^2)(\text{x}^2+\text{b}^2)}\ \text{dx}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{1}{1+\sqrt{\tan\text{x}}}\text{ dx}$

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

$\text{If A} = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \text{find A}^{2} - \text{5 A + 4I} $ and hence find a matrix X such that $\text{A}^{2} - \text{5A + 4I + X = 0}$

If a, b, c are real numbers such that $\begin{vmatrix}\text{b}+\text{c}&\text{c}+\text{a}&\text{a}+\text{b}\\\text{c}+\text{a}&\text{a}+\text{b}&\text{b}+\text{c}\\\text{a}+\text{b}&\text{b}+\text{c}&\text{c}+\text{a}\end{vmatrix}=0,$ then show that either a + b + c = 0 or a = b= c.

Evaluate the follwing intregals:
$\int\frac{2\text{x}}{(\text{x}^2+1)(\text{x}^2+2)^2}\ \text{dx}$

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}-\text{x}\cos^2\Big(\frac{\text{y}}{\text{x}}\Big)$

Verify Rolle's theorem for the following function on the indicated intervals
f(x) = x² - 8x + 12 on [2, 6]

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

If $\text{a}(1-\cos\theta),\text{y}=\text{a}(\theta+\sin\theta),$ prove that, $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{1}{\text{a}}$ at $\theta=\frac{\pi}{2}.$