Solve the following differential equations: (x-1) dy dx =2x^3y - Vidyadip

Question

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

✓

Answer

We have,
$(\text{x}-1)\frac{\text{dy}}{\text{dx}}=2\text{x}^3\text{y}$
$\Rightarrow\frac{1}{\text{y}}\text{dy}=\frac{2\text{x}^3}{\text{x}-1}\text{dx}$
Integrating both sides, we get
$\int\frac{1}{\text{y}}\text{dy}=\int\frac{2\text{x}^3}{\text{x}-1}\text{dx}$
$\Rightarrow\log|\text{y}|=2\int\frac{\text{x}^3-1+1}{\text{x}-1}\text{dx}$
$\Rightarrow\log|\text{y}|=2\Big[\int\frac{\text{x}^3-1}{\text{x}-1}\text{dx}+\int\frac{1}{\text{x}-1}\text{dx}\Big]$
$\Rightarrow\log|\text{y}|=2\Big[\int\frac{(\text{x}-1)(\text{x}^2+\text{x}+1)}{\text{x}-1}\text{dx}+\int\frac{1}{\text{x}-1}\text{dx}\Big]$
$\Rightarrow\log|\text{y}|=2\Big[\int(\text{x}^2+\text{x}+1)\text{dx}+\int\frac{1}{\text{x}-1}\text{dx}\Big]$
$\Rightarrow\log|\text{y}|=2\Big[\frac{\text{x}^3}{3}+\frac{\text{x}^2}{2}+\text{x}+\log|\text{x}-1|\Big]+\text{C}$
$\Rightarrow\log|\text{y}|=\frac{2}{3}\text{x}^3+\text{x}^2+2\text{x}+\log|\text{x}-1|^2+\text{C}$
$\Rightarrow\text{y}=\text{e}^{\frac{2}{3}}\text{x}^3+\text{x}^2+2\text{x}+\log|\text{x}-1|^2+\text{C}$
$\Rightarrow\text{y}=\text{e}^{\text{C}}\times\text{e}^{\log|\text{x}-1|^2}\times\text{e}^{\frac{2}{3}}\text{x}^3+\text{x}^2+2\text{x}$
$\Rightarrow\text{y = C}_1|\text{x}-1|^2\text{e}^{\frac{2}{3}}\text{x}^3+\text{x}^2+2\text{x}$ $\big[\because\text{e}^{\text{In x}}=\text{x}\text{and where, C}_1=\text{e}^{\text{c}}\big]$
$\therefore\text{y}=\text{C}_1|\text{x}-1|^2\text{e}^{\frac{2}{3}}\text{x}^3+\text{x}^2+2\text{x}$ is 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 "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

A manufacturer makes two products A and B. Product A sells at Rs. 200 each and takes 1/2 hour to make. Product B sells at Rs. 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs 10000. If the profit on each of product A is Rs. 20 and on product B is Rs. 30, then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.

Evalute the following integrals:
$\int\frac{1-\cot\text{x}}{1+\cot\text{x}}\text{dx}$

Find the $n^{\text {th }}$ derivative of the following : $e^{a x} \sin (b x+c)$

Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_0\frac{\tan^{7}\text{x}}{\tan^{7}\text{x}+\cot^7\text{x}}\text{ dx}$

A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.

Evaluate:
$\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$

The vertices A, B, C of triangle ABC have respectively position vector $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ with respect to given origin O. Show that the point D where the bisector of $\angle{\text{A}}$ meets BC has position Vector $\vec{\text{d}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$, where $\beta=\big|\vec{\text{c}}-\vec{\text{a}}\big|$ and, $\gamma=\big|\vec{\text{a}}-\vec{\text{b}}\big|$.

If $\text{y}=\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}},$ prove that $2\text{x}\frac{\text{dy}}{\text{dx}}=\sqrt{\text{x}}-\frac{1}{\sqrt{\text{x}}}$

Show that the relation R on the set A = {x ∈ Z; 0 ≤ x ≤ 12}, given by R = {(a, b): a = b}, is an equivalence relation. Find the set of all elements related to 1.

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs 100,000 and each flight of a model 535 plane costs Rs 150,000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.