Solve the following differential equation: dy dx + y x =x^3 - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{x}^3$

✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{x}^3\ \dots(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
where
$\text{P}=-\frac{1}{\text{x}}$
$\text{Q}=\text{x}^3$
$\therefore$ I.F. $=\text{e}^{\int\text{Pdx}}=\text{e}^{\int\frac{1}{\text{x}}\text{dx}}$
$\text{e}^{\log|\text{x}|}=\text{x}$
Multiplying both sides of (1) by x, we get
$\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\frac{1}{\text{x}}\text{y}\Big)=\text{x x}^3$
$\Rightarrow\ \text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{x}^4$
Integrating both sides with respect to x, we get
$\text{xy}=\int\text{x}^4\text{dx + C}$
$\Rightarrow\ \text{xy}=\frac{\text{x}^5}{5}+\text{C}$
$\Rightarrow\ 5\text{xy}=\text{x}^5+5\text{C}$
$\Rightarrow\ 5\text{xy}=\text{x}^5+\text{K}$ (where, K = 5C)
Hence, $5\text{xy}=\text{x}^5+\text{K}$ 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 "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 = 3x + 3y, if possible,
Subject to the constraints
$\text{x}-\text{y}\leq1$
$\text{x}+\text{y}\geq3$
$\text{x},\text{y}\geq0$

Differentiate the following functions with respect to x:
$\log\Big(\frac{\sin\text{x}}{1+\cos\text{x}}\Big)$

Find the particular solution of the differential equation $(1 + y^2) + (x – \text{e}^{\tan^{-1}}y)\frac{\text{dy}}{\text{dx}}=0$ given that $y = 0$ when $x=1.$

Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is $\text{y}+2(\text{x}+1)=2\text{e}^{2\text{x}}.$

A curve is such that the length of the perpendicular from the origin on the tangent at any point P of the curve is equal to the abscissa of P. Prove that the differential equation of the curve is $\text{y}^{2}-2\text{xy}\frac{\text{dy}}{\text{dx}}-\text{x}^{2}=0$ and hence find the curve.

One bag contains $4$ yellow and $5$ red balls. Another bag contains $6$ yellow and $3$ red balls. $A$ ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find the probability that ball drawn is yellow.

Using differentials, find the approximate values of the following:
$(29)^{\frac{1}{3}}$

Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$

If $e^y= y^x$, prove that $\frac{\text{dy}}{\text{dx}}=\frac{(\log\text{y})^2}{\log\text{y}-1}$

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