Solve the following differential equations: dy dx =2e^ x y^3,y(0)… - Vidyadip

Question

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=2\text{e}^{\text{x}}\text{y}^3,\text{y}(0)=\frac{1}{2}$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}=2\text{e}^{\text{x}}\text{y}^3,\text{y}(0)=\frac{1}{2}$
$\Rightarrow\frac{1}{\text{y}^3}\text{dy}=2\text{e}^{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{1}{\text{y}^3}\text{dy}=\int2\text{e}^{\text{x}}\text{dx}$
$\Rightarrow-\frac{1}{2\text{y}^3}=2\text{e}^{\text{x}}+\text{C}...(1)$
Given: at $\text{x}=0,\text{y}=\frac{1}{2}$
Substituting the valuse of x and y in (1), we get
$-\frac{1}{2\times\frac{1}{4}}=2\text{e}^{0}+\text{C}$
$\Rightarrow\text{C}=-2-2$
$\Rightarrow\text{C}=-4$
Substituting the value of C in (1), we get
$\Rightarrow-\frac{1}{2\text{y}^2}=2\text{e}^{\text{x}}-4$
$\Rightarrow\text{y}^{2}(8-4\text{e}^{\text{x}})=1$
Hence, $\text{y}^{\text{x}}(8-4\text{e}^{\text{x}})=1$ is the required solution.

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

$\text{Let}\ \vec{\text{a}}=4\hat{\text{i}}+5\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{c}}\ \text{and }\vec{\text{b}}\ \text{and}\ \vec{\text{d}}\cdot\vec{\text{a}}=21.$

Find the shortest distance between the lines
$\vec{\text{r}}=\Big(4\hat{\text{i}}-\hat{\text{j}}\Big)+\lambda\Big(\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)$ $\text{and}\ \vec{\text{r}}=\Big(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}\Big)+\mu\Big(2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}\Big).$

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

Three schools A, B and C organised a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of ₹ 25, ₹ 100 and ₹ 50 each. The number of articles sold are given below:

School	A	B	C
Article	A	B	C
Hand - fans	40	25	35
Mats	50	40	50
Plates	20	30	40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation.

Evaluvate the following intregals:
$\int\frac{1}{1-\tan\text{x}}\text{ dx}$

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^2+\text{x}}\text{dx}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
y³ - 3xy² = x³ + 3x²y

Evaluate the following integrals as limit of sum:
$\int\limits^2_{1}\big(\text{x}^2-1\big)\text{dx}$

If a, b and c are real numbers, and
$\triangle=\begin{vmatrix}b+c&c+a&a+b\\c+a&a+b&b+c\\a+b&b+c&c+a\end{vmatrix}=0,$
Show that either a + b + c = 0 or a = b = c.

If $\text{x}=\text{e}^{\cos2\text{t}}$ and $\text{y}=\text{e}^{\sin2\text{t}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$