Download App

Differential Equations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferential Equations5 Marks
Question
Solve the following initial value problems:
$(\text{y}^4-2\text{x}^3\text{y})\text{dx}+(\text{x}^4-2\text{xy}^3)\text{dy}=0,\text{y}(1)=1$
✓

Answer

$(\text{y}^4-2\text{x}^3\text{y})\text{dx}+(\text{x}^4-2\text{xy}^3)\text{dy}=0,\text{y}(1)=1$
This is a homogeneous equation, put y = vx
$\big(\text{v}^4\text{x}^4-2\text{vx}^4\big)+\big(\text{x}^4-2\text{v}^3\text{x}^4\big)\Big[\text{v + x}\frac{\text{dv}}{\text{dx}}\Big]=0$
$\big(\text{v}^4\text{x}^4-2\text{vx}^4\big)=\big(2\text{v}^3\text{x}^4-\text{x}^4\big)\Big[\text{v + x}\frac{\text{dv}}{\text{dx}}\Big]$
$\text{vx}^4(\text{v}^3-2)=\text{x}^4(2\text{v}^3-1)\Big[\text{v + x}\frac{\text{dv}}{\text{dx}}\Big]$
$\text{v}(\text{v}^3-2)=(2\text{v}^3-1)\text{v + x}(2\text{v}^3-1)\frac{\text{dv}}{\text{dx}}$
$\text{v}\big[\text{v}^3-2-2\text{v}^3+1\big]=\text{x}(2\text{v}^3-1)\frac{\text{dv}}{\text{dx}}$
$\text{v}(-1-\text{v}^3)=\text{x}(2\text{v}^3-1)\frac{\text{dv}}{\text{dx}}$
$\text{v}(1+\text{v}^3)=\text{x}(1-2\text{v}^3)\frac{\text{dv}}{\text{dx}}$
$\frac{\text{dx}}{\text{x}}=\frac{(1-2\text{v}^3)}{\text{v}(1+\text{v}^3)}\text{dv}$
On integrating both side of the equation we get,
$\int\frac{\text{dx}}{\text{x}}=\int\frac{(1-2\text{v}^3)}{\text{v}(1+\text{v}^3)}\text{dv}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\int\frac{1+\text{v}^3-3\text{v}^3}{\text{v}(1+\text{v}^3)}\text{dv}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\int\frac{1+\text{v}^3}{\text{v}(1+\text{v}^3)}\text{dv}-\int\frac{3\text{v}}{\text{v}(1+\text{v}^3)}\text{dv}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\int\frac{1}{\text{v}}\text{dv}-\int\frac{3\text{v}^2}{(1+\text{v}^3)}\text{dv}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\log_{\text{e}}\text{v}-\int\frac{\text{dt}}{\text{t}}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\log_{\text{e}}\text{v}-\log_{\text{e}}(1+\text{v}^3)+\text{C}$ let $(1+\text{v}^3)=\text{t},3\text{v}^2\text{dv}=\text{dt}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\log_{\text{e}}\frac{\text{v}}{1+\text{v}^3}+\text{C}$
As $\text{v}=\frac{\text{y}}{\text{x}}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\log_{\text{e}}\frac{\frac{\text{y}}{\text{x}}}{1+\text{y}^{\frac{3}{\text{x}}}}+\text{C}$
$\Rightarrow\ \log_{\text{e}}\text{x}=\log_{\text{e}}\frac{\text{yx}^2}{\text{x}^3+\text{y}^3}+\text{C}$
As y(1) = 1
$\Rightarrow\ \log_{\text{e}}1=\log_{\text{e}}\frac{1}{1+1}+\text{C}$
$\Rightarrow\ 0=\log_{\text{e}}\frac{1}2+\text{C}$
$\text{C}=-\log_{\text{e}}\frac{1}2$
$\Rightarrow\ \text{C}=\log_{\text{e}}2$
$\therefore\ \log_{\text{e}}\text{x}=\log_{\text{e}}\frac{\text{yx}^2}{\text{x}^3+\text{y}^3}+\log_{\text{e}}2$

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

Maximize $Z = 9x + 3y$
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
→
Evaluate the following integrals:$\int\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)\text{dx}$
→
Differentiate the following functions with respect to x:
$\text{e}^{\tan^{-1}\sqrt{\text{x}}}$
→
If $x^m + y^n = 1,$ Prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$
→
A fair coin is tossed four times. Let X denote the longest string of heads accuring. Find the probability distribution mean and variance of X.
→
Solve the following systems of homogeneous linear equations by matrix method:
2x + 3y - z = 0
x - y - 2z = 0
3x + y + 3z = 0
→
If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=0,$ then show that $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{b}}\times\vec{\text{c}}=\vec{\text{c}}\times\vec{\text{a}}.$ Interpret the result geometrically?
→
At what points on the following curves, is the tangent parallel to $x-$axis$?$
$y = 12(x + 1)(x - 2)$ on $[-1, 2]$
→
Evaluate the following integrals:
$\int\cos^3\sqrt{\text{x}}\text{dx}$
→
Find the shortest distance between the lines
$\frac{\text{x}+1}{7}=\frac{\text{y}+1}{-6}=\frac{\text{z}+1}{1}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}-5}{-2}=\frac{\text{z}-7}{1}$
→