Solve the following differential equation dy dx =x^5 ^ -1 (x^3) - Vidyadip

Question

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{x}^5\tan^{-1}(\text{x}^3)$

✓

Answer

$\frac{\text{dy}}{\text{dx}}=\text{x}^5\tan^{-1}(\text{x}^3)$$\text{dy}=\text{x}^5\tan^{-1}(\text{x}^3)\text{dx}$
$\int\text{dy}=\int\text{x}^5\tan^{-1}(\text{x}^3)\text{dx}$
put $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^2\text{dx}=\text{dt}$
$\Rightarrow\text{x}^2\text{dx}=\frac{\text{dt}}{3}$
So,
$\int\text{dy}=\frac{1}{3}\Big[\tan^{-1}\text{t}\int\text{t dt}=\int\Big(\frac{1}{1+\text{t}^2}\Big)\times\int\text{t dx}\Big)\Big]\text{dt}+\text{C}$
Using integration by parts
$\text{y}=\frac{1}{3}\Big[\frac{\text{t}^2}{2}+\tan^{-1}-\int\frac{\text{t}^2}{2(\text{t}^2+1)}\text{dt}\Big]+\text{C}$
$=\frac{1}{6}\text{t}^2\tan^{-1}\text{t}-\frac{1}{6}\int\Big(\frac{\text{t}^2}{\text{t}^2+1}\Big)\text{dt}+\text{C}$
$\text{y}=\frac{1}{6}\text{t}^2\tan^{-1}\text{t}-\frac{1}{6}\int\Big(1-\frac{1}{\text{t}^2+1}\Big)\text{dt}+\text{C}$
$=\frac{1}{6}\text{t}^2\tan^{-1}\text{t}-\frac{1}{6}\text{t}+\frac{1}{6}\tan^{-1}\text{t}+\text{C}$
$\text{y}=\frac{1}{6}(\text{t}^2+1)\tan^{-1}\text{t}-\frac{1}{6}\text{t}+\text{C}$
$\text{y}=\frac{1}{6}[(\text{t}^2+1)\tan^{-1}\text{t}-\text{t}]+\text{C}$
So,
$\text{y}=\frac{1}{6}[(\text{x}^6+1)\tan^{-1}(\text{x}^3)-\text{x}^3]+\text{C}$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that the curves $y^2 = 4x$ and $x^2 + y^2 - 6x + 1 = 0$ touch each other at the point $(1, 2).$

$\int\limits^\pi_0\frac{\text{x}}{1+\sin\text{x}}$

Show that the lines
$\vec{\text{r}}=3\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=5\hat{\text{i}}-2\hat{\text{j}}+\mu\big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\big)$ are intersecting. Hence, find their point of intersection.

show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^2}$ is a solution of the differential equation $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$

Evaluate the following intregals:
$\int\frac{\text{x}^3}{(\text{x}-1)(\text{x}-2)(\text{x}-3)}\ \text{dx}$

Solve the following differential equation:
$(2\text{x}-10\text{y}^3)\frac{\text{dx}}{\text{dy}}+\text{y}=0$

A factory makes tennis rackets and cricket bats.A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and crickets bats that the factory must manufacture to earn the maximum profit.Make it as an L.P.P. and solve graphically.

Maximize Z = 50x + 30y
Subject to
$2\text{x}+\text{y}\leq18$
$3\text{x}+2\text{y}\leq34$
$\text{x},\text{y}\geq0$

A and B are two events such that $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{4}.$ Find:

$\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$
$\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$
$\text{P}\Big(\frac{\text{A}'}{\text{B}}\Big)$
$\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)$

Show that the following system of linear equations is consistent and also find solutions:
$5x +3y + 7z = 4$
$3x + 26y + 2z = 9$
$7x + 2y + 10z = 5$