Solve the following differential equations: dy dx =2e^ 2x y^2,y(0… - Vidyadip

Question

Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=2\text{e}^{2\text{x}}\text{y}^2,\text{y}(0)=-1$

✓

Answer

$\frac{\text{dy}}{\text{dx}}=2\text{e}^{2\text{x}}\text{y}^2,\text{y}(0)=-1$
$\Rightarrow\frac{1}{\text{y}^2}\text{dy}=2\text{e}^{2\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{1}{\text{y}^2}\text{dy}=2\int\text{e}^{2\text{x}}\text{dx}$
$\Rightarrow\frac{-1}{\text{y}}=\text{e}^{2\text{x}}+\text{C}...(1)$
We know that at $\text{x}=0,\text{y}=-1.$
Substituting the values of x and y in (1), we get
$1=1+\text{C}$
$\Rightarrow\text{C}=0$
Substituting the value of C in (1), we get
$-\frac{1}{\text{y}}=\text{e}^{2\text{x}}$
$\Rightarrow\text{y}=-\text{e}^{-2\text{x}}$
Hence, $\text{y}=-\text{e}^{-2\text{x}}$ is the required soluton.

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

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}},&\text{x}<\frac{{\pi}}{2}\\3,&\text{x}=\frac{\pi}{2}\\\frac{3\tan\text{x}}{2\text{x}-\pi},&\text{x}>\frac{\pi}{2}\end{cases}$

Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}})$ and the plane $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5.$

$A$ bag $A$ contains $5$ white and $6$ black balls. Another bag $B$ contains $4$ white and $3$ black balls. $A$ ball is transferred from bag $A$ to the bag $B$ and then a ball is taken out of the second bag. Find the probability of this ball being black.

$\overrightarrow{A B}=3 \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{C D}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ are two vectors. The position vectors of the points $A$ and $C$ are $6 \hat{i}+7 \hat{j}+4 \hat{k}$ and $-9 \hat{j}+2 \hat{k}$, respectively. Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{P Q}$ is perpendicular to $\overrightarrow{A B}$ and $\overrightarrow{C D}$ both.

Given $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1\end{array}\right]$ find $AB$ and use this result in solving the following system of equations.
$x - y + z =4$
$x - 2y - 2z = 9$
$2x + y + 3z = 1$

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

Evaluate the following integrals:$\int\frac{(3\sin\text{x}-2)\cos\text{x}}{5-\cos^2\text{x}-4\sin\text{x}}\text{ dx}$

Evaluate:$\lim\limits_{\text{y|} \to \infty}\Big(\sqrt{\text{x}^{2}+\text{x + 1}}-\text{x}\Big).$

Prove that the line of section of the planes $5x + 2y - 4z + 2 = 0$ and $2x + 8y + 2z - 1 = 0$ is parallel to the plane $4x - 2y - 5z - 2 = 0.$

The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.