Download App

Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=2\text{x}+\text{x}^2\tan\text{x},\text{ y}(0)=1$
✓

Answer

We have,
$\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=2\text{x}+\text{x}^2\tan\text{x}\ ...(1)$
Clearly, it is a linear differential equation of the form
$\frac{\text{dy}}{\text{dx}}+\text{Px}=\text{Q}$
Where $\text{P}=\tan\text{x}$ and $\text{Q}=\text{x}^2\cot\text{x}+2\text{x}$
$\therefore\text{ I.F.}=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\tan\text{x dx}}$
$=\text{e}^{\log|\sec\text{x}|}$
$=\sec\text{x}$
Multiplying both sides of (1) by $\text{I.F.}=\sec\text{x},$ we get
$\sec\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}\Big)=\sec\text{x}(\text{x}^2\tan\text{x}+2\text{x})$
$\Rightarrow\sec\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}\Big)=\text{x}^2\tan\text{x }\sec\text{x}+2\text{x}\sec\text{x}$
Integrating both sides with respect to x, we get
$\text{y}\sec\text{x}=\int\text{x}^2\tan\text{x }\sec\text{x dx}+2\int\text{x}\sec\text{x dx}\\+2\int\text{x}\sec\text{x dx} +\text{C}$
$\Rightarrow\text{y}\sec\text{x}=\int\text{x}^2\tan\text{x }\sec\text{x dx}+2\sec\text{x}\int\text{x dx}\\-2\int\Big[\frac{\text{d}}{\text{dx}}(\sec\text{x})\int\text{x dx}\Big]\text{dx}+\text{C}$
$\Rightarrow\text{y}\sec\text{x}=\int\text{x}^2\tan\text{x }\sec\text{x dx}+\text{x}^2\sec\text{x}\\-\int\text{x}^2\tan\text{x }\sec\text{x dx} +\text{C}$
$\Rightarrow\text{y}\sec\text{x}=\text{x}^2\sec\text{x}+\text{C}$
$\Rightarrow\text{y}=\text{x}^2+\text{C}\cos\text{x}\ ...(2)$
Now,
$\text{y}(0)=1$
$\therefore\ 1=0+\text{C}\cos0$
$\Rightarrow\text{C}=1$
Putting the value of C in (2), we get
$\text{y}=\text{x}^2+\cos\text{x}$
Hence, $\text{y}=\text{x}^2+\cos\text{x}$ 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 "4 Marks" 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

Let $\text{A} = \text{R} - \left\{3\right\}, \text{B} = \text{R} - \left\{1\right\}. \text{Let f : A} \rightarrow \text{B}$ be defined by $\text{f(x)}\frac{\text{x - 2}}{\text{x - 3}}, \forall \text{ x} \in \text{A}.$ Show that f is bijective. Also, find
  1. $\text{x, if f}^{-1} \text{(x) = 4}$
  2. $\text{f}^{-1} (7)$
→
If $\text{y}\sin(\text{x}^\text{x}),$ prove that $\frac{\text{dy}}{\text{dx}}=\cos(\text{x}^\text{x})\times\text{x}^\text{x}(1+\log\text{x})$
→
If y = tan-1 $\Bigg[\frac{\sqrt{1+\text{x}^{2}}-\sqrt{1-\text{x}^{2}}}{\sqrt{1+\text{x}^{2}}+\sqrt{1-\text{x}^{2}}}\Bigg],\text{ find }\frac{\text{dy}}{\text{dx}}.$
→
Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?
→
Find the second order derivatives of the following functions:
$\text{y}=\tan^{-1}\text{x}$
→
Tangent to the circle $\text{x}^{2} + \text{y}^{2} = 4$ at any point on it in the first quadrant makes intercepts OA and OB on x and y axes respectively, O being the centre of the circle. Find the minimum value of (OA + OB).
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\cos^3\text{x}\sin^2\text{x}+\text{x}\sqrt{2\text{x}+1}$
→
Find the particular solution of the differential equation satisfying the given conditions:
x2dy + (xy + y2) dx = 0 ; y = 1 when x = 1.
→
Differentiate the following functions with respect to x:
$(\text{x}^\text{x})\sqrt{\text{x}}$
→
A gardener has supply of fertilizer of type I which consists of 10% nitrogen and 6% phosphoric acid and type II fertilizer which consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14kg of nitrogen and 14kg of phosphoric acid for his crop. If the type I fertilizer costs 60 paise per kg and type II fertilizer costs 40 paise per kg, determine how many kilograms of each fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
→