Solve the following differential equations : xy d y d x = x ^2+2 … - Vidyadip

Question

Solve the following differential equations : $xy \frac{d y}{d x}= x ^2+2 y ^2$

✓

Answer

$
\begin{array}{r}
x y \frac{d y}{d x}=x^2+2 y^2 \\
\therefore \frac{d y}{d x}=\frac{x^2+2 y^2}{x y}
\end{array}
$
Put $y=v x$. Then $\frac{d y}{d x}=v+x \frac{d v}{d x}$
$\therefore$ (1) becomes, $v+x \frac{d v}{d x}=\frac{x^2+2 v^2 x^2}{x \cdot v x}=\frac{1+2 v^2}{v}$
$
\therefore x \frac{d v}{d x}=\frac{1+2 v^2}{v}-v=\frac{1+2 v^2-v^2}{v}
$
$\therefore x \frac{d v}{d x}=1+v^2$
$\therefore x \frac{d v}{d x}=\frac{1+v^2}{v}$
$
\therefore \frac{v}{1+v^2} d v=\frac{1}{x} d x
$
Integrating, we get
$
\begin{aligned}
& \int \frac{v}{1+v^2} d v=\int \frac{1}{x} d x \\
\therefore & \frac{1}{2} \int \frac{2 v}{1+v^2} d v=\int \frac{1}{x} d x+\log c_1 \\
\therefore & \frac{1}{2} \log \left|1+v^2\right|=\log |x|+\log c_1 \\
\therefore & \log \left|1+v^2\right|=2 \log |x|+2 \log c_1 \\
\therefore & \log \left|1+v^2\right|=\log \left|x^2\right|+\log c_1{ }^2 \\
\therefore & \log \left|1+v^2\right|=\log \left|c x^2\right|, \text { where } c=c_1{ }^2 \\
\therefore & 1+v^2=c x^2
\end{aligned}
$
$
\begin{aligned}
& \therefore 1+\frac{y^2}{x^2}=c x^2 \\
& \therefore \frac{x^2+y^2}{x^2}=c x^2 \\
& \therefore x^2+y^2=c x^4
\end{aligned}
$
This is the general 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 Equation and Applications (p-1)→Differential Equation and Applications (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

If $a x^2+2 h x y+b y^2=0$, then show that $\frac{d^2 y}{d x^2}=0$.

Find the inverse of $\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7\end{array}\right]$ by the elementary row transformations.

Evalute : $\int \frac{3 x-1}{2 x^2-x-1} d x$

Assuming the first statement as p and second as q, write the following statements in symbolic form:
(i) $x^3 + y^3 = (x + y)^3$^, iff $xy = 0.$
(ii) The drug is effective though it has side effects.
(iii) If a real number is not rational, then it must be irrational.
(iv) It is not true that Ram is tall and handsome.
(v) Even though it is not cloudy, it is still raining.
(vi) It is not true that intelligent persons are neither polite nor helpful.
(vii) If the question paper is not easy, then we shall not pass.

The sum of the cost of one Economics book, one Cooperation book, and one Account book is $₹ 420$. The total cost of an Economic book, $2$ Cooperation books, and an Account book is $₹ 480$. Also, the total cost of an Economic book, $3$ Cooperation books, and $2$ Account books is $₹ 600$. Find the cost of each book.

Find the inverse of $\left[\begin{array}{lll}3 & 1 & 5 \\ 2 & 7 & 8 \\ 1 & 2 & 5\end{array}\right]$ by adjoint method.

A bill drawn on $5$th June for $6$ months was discounted at the rate of $5\%$ p.a. on $19$th October. If the cash value of the bill is $\text{₹} 43,500,$ find the face value of the bill.

Solve the following equations by the method of inversion : $x + y + z = 1, x – y + z = 2$ and $x + y – z = 3$

Solve the following differential equations : $x \frac{d y}{d x}+2 y=x^2 \cdot \log x$

Obtain trend values for data in Problem 4 using 4-yearly centered moving averages.