Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS4 Marks
Question
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}=\cos\text{x}$
✓

Answer

Here, $\frac{\text{dy}}{\text{dx}}+\text{y}=\cos\text{x}$
It is a linear differential equation. Comparing the equation by,
$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=1,\text{Q}=\cos\text{x}$
I.F. $=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\text{dx}}$
$=\text{e}^{\text{x}}$
Solution of the equation is given by,
$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$
$\text{y}(\text{e}^{\text{x}})=\int(\cos\text{x})(\text{e}^{\text{x}})+\text{C}_1\ \dots(\text{i})$
Let $\text{I}=\int\text{e}^{\text{x}}\cos\text{xdx}$
$=\cos\text{x}\times\int\text{e}^{\text{x}}\text{dx}\int(\sin\text{x}\int\text{e}^{\text{x}}\text{dx})\text{dx + C}_2$
Using integration by parts
$\text{I}=\text{e}^{\text{x}}\cos{\text{x}}+\int\sin\text{xe}^{\text{x}}\text{dx + C}$
$=\text{e}^{\text{x}}\cos\text{x}+\big[\sin\text{x}\int\text{e}^{\text{x}}\text{dx}-\int(\cos\text{x}\int\text{e}^{\text{x}}\text{dx})\text{dx}\big]+\text{C}_2$
$\text{I}=\text{e}^{\text{x}}\cos\text{x}+\sin\text{e}^\text{x}-\text{I}+\text{C}_2$
$2\text{I}=\text{e}^{\text{x}}(\cos\text{x}+\sin\text{x})+\text{C}_2$
$\text{I}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin\text{x})+\frac{\text{C}_2}{2}$
$\text{I}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin\text{x})+\text{C}_3$
Putting I in equation (i),
$\text{ye}^{\text{x}}=\frac{\text{e}^{\text{x}}}{2}(\cos\text{x}+\sin{\text{x}})+\text{C}_1+\text{C}_3$
$\text{ye}^{\text{x}}=\frac{\text{e}^{\text{x}}}2(\cos\text{x}+\sin\text{x})+\text{C}$
$\text{y}=\frac{1}2(\cos\text{x}+\sin\text{x})+\text{Ce}^{-\text{x}}$

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

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs 17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?
→
Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\Big(\sqrt{\tan\text{x}}+\sqrt{\cot}\text{x}\Big)\text{dx}$
→
Find the equation of the lines joining the following pairs of vertices and then find the shortest distance between the lines

(1) (0, 0, 0) and (1, 0, 2)

(2) (1, 3, 0) and (0, 3, 0)

→
Differentiate the following with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big)$
→
Using differentials, find the approximate values of the following:
$\cos61^\circ$ it being given that $\sin60^\circ=0.86603$ and $0.01745$ radian
→
If $\text{f(x)}=\begin{cases}\frac{\cos^2\text{x}-\sin^2\text{x}}{\sqrt{\text{x}^2+1}-1},&\text{x}\neq0\\\text{k},&\text{x}=0\end{cases}$ is continuous at x = 0, find k.
→
A circular metal plate expends under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10cm.
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{ dx}$
→
The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.
→
A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
$\text{M}=\frac{\text{WL}}{2}\text{x}-\frac{\text{W}}{2}\text{x}^{2}$
Find the point at which M is maximum in each case.
→