# Find One-parameter Families of Solution Curves of the Following Differential Equation:- X D Y D X + 2 Y = X 2 Log X - Mathematics

Sum

Find one-parameter families of solution curves of the following differential equation:-

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

Solve the following differential equation:-

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

#### Solution

We have,
$x\frac{dy}{dx} + 2y = x^2 \log x$
Dividing both sides by x, we get
$\frac{dy}{dx} + \frac{2y}{x} = x \log x$
$\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get}$
$P = \frac{2}{x}$
$Q = x \log x$
Now,
$I . F . = e^{\int P\ dx} = e^{\int\frac{2}{x}dx}$
$= e^{2\log\left| x \right|}$
$= x^2$
So, the solution is given by
$y \times I . F . = \int Q \times I . F . dx + C$

$\Rightarrow x^2 y = \log x\int x^3 dx - \int\left[ \frac{d}{dx}\left( \log x \right)\int x^3 dx \right]dx + C$
$\Rightarrow x^2 y = \frac{x^4 \log x}{4} - \int\frac{x^3}{4}dx + C$
$\Rightarrow x^2 y = \frac{x^4 \log x}{4} - \frac{x^4}{16} + C$
$\Rightarrow y = \frac{x^2 \log x}{4} - \frac{x^2}{16} + \frac{C}{x^2}$
$\Rightarrow y = \frac{x^2}{16}\left( 4\log x - 1 \right) + \frac{C}{x^2}$

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 22 Differential Equations
Exercise 22.1 | Q 36.12 | Page 107