Solve the following differential equation: x dy=(2y+2x^4+x^2)dx - Vidyadip

Question

Solve the following differential equation:
$\text{x dy}=(2\text{y}+2\text{x}^4+\text{x}^2)\text{dx}$

✓

Answer

Here, $\text{x dy}=(2\text{y}+2\text{x}^4+\text{x}^2)\text{dx}$
$\text{x}\frac{\text{dy}}{\text{dx}}=2\text{y}+2\text{x}^4+\text{x}^2$
$\frac{\text{dy}}{\text{dx}}-\frac{2}{\text{x}}\text{y}=2\text{x}^3+\text{x}$
It is a linear differential equation. Comparing it with equation,
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=-\frac{2}{\text{x}},\text{Q}=2\text{x}^3+\text{x}$
I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{-2\int\frac{1}{1+\text{x}}\text{dx}}$
$=\text{e}^{-2\log|\text{x}|}$
$=\text{e}^{\log\big(\frac{1}{\text{x}^2}\big)}$
$=\frac{1}{\text{x}^2}$
Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$
$\text{y}\Big(\frac{1}{\text{x}^2}\Big)=\int\big(2\text{x}^3+\text{x}\big)\Big(\frac{1}{\text{x}^2}\Big)\text{dx + C}$
$\frac{\text{y}}{\text{x}^2}=\int\Big(2\text{x}+\frac{1}{\text{x}}\Big)\text{dx + C}$
$\frac{\text{y}}{\text{x}^2}=2\frac{\text{x}^2}2+\log|\text{x}|+\text{C}$
$\text{y}=\text{x}^4+\text{x}^2\log|\text{x}|+\text{Cx}^2$

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

Find the area of the region enclosed by the parabola $x^2 = y$ and the line $y = x + 2.$

Determine if f defined by:
$\text{f(x)}=\begin{cases}\text{x}^{2} \sin\frac{1}{\text{x}}, \text{if} \ \text{x}\neq0\\0, \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \text{x} = 0\end{cases}$

If y(t) is a solution of $(1+\text{t})\frac{\text{dy}}{\text{dt}}-\text{ty}=1$ and y(0) = -1, then show that $\text{y}(1)=-\frac{1}{2}.$

Evaluate the following definite integrals:
$\int_{\text{e}}^\limits{\text{e}^2}\Big\{\frac{1}{\log\text{x}}-\frac{1}{(\log\text{x})^2}\Big\}\text{dx}$

If $\text{x}^\text{y}-\text{y}^\text{x}=\text{a}^\text{b},$ find $=\frac{\text{dy}}{\text{dx}}.$

If $\text{x}\sin(\text{a}+\text{y})+\sin\text{a}\cos(\text{a}+\text{y})=0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin\text{a}}$

A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him ₹ 360 and a manually operated sewing machine ₹ 240. He can sell the sewing machine at a profit of ₹ 22 and a manually operated sewing machine at a profit of ₹ 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically.

Evaluate the following integrals:
$\int\text{e}^{2\text{x}}\sin\text{x}\cos\text{x }\text{dx}$

A man 180cm tall walks at a rate of 2m/ sec. away, from a source of light that is 9m above the ground. How fast is the length of his shadow increasing when he is 3m away from the base of light?

If $B, C$ are n rowed square matrices and if $A = B + C, BC = CB, C^2 = O,$ then show that for every $n \in N, A^{n+1} = B^n(B + (n + 1)C).$