Download App

DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 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 "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

A ladder $5$ m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2$ cm/s. How fast is its height on the wall decreasing when the foot of the ladder is $4$ m away from the wall?
→
Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
→
Prove that the curves $x = y^2 $ and $xy = k$ cut at right angles if $8k^2 = 1.$
→
Find the area of the region enclosed between the two curve $x^2 + y^2 = 9$ and $(x - 3)^2 + y^2 = 9$.
→
If $\vec{\text{a}} \times \vec{\text{b}} = \vec{\text{c}} \times \vec{\text{d}}$ and $\vec{\text{a}} \times \vec{\text{c}} = \vec{\text{b}} \times \vec{\text{d}},$ show that $\vec{\text{a}} - \vec{\text{d}}$ is parallel to $\vec{\text{b}} - \vec{\text{c}},$ where $\vec{\text{a}} \neq \vec{\text{d}}$ and $\vec{\text{b}} \neq \vec{\text{c}}.$
→
If w is a complex cube root of unity, show that.
$\begin{pmatrix}\begin{bmatrix}1&w&w^2\\w&w^2&1\\w^2&1&w\end{bmatrix} +\begin{bmatrix}w&w^2&1\\w^2&1&w\\w&w^2&1\end{bmatrix}\end{pmatrix}\begin{bmatrix}1\\w\\w^2\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$
→
If $B, C$ are $n$ rowed square matrices and if $A = B + C, BC = CB, C^2 = O$, then show that for every $n \in N, A^{n+1} = B^n(B + (n + 1)C).$
→
Solve the following differential equations:
$\text{xy}\frac{\text{dy}}{\text{dx}}=(\text{x}+2)(\text{y}+2),\text{y}(1)=-1$
→
$\triangle=\begin{vmatrix}\cos\alpha\cos\beta&\cos\alpha\sin\beta&-\sin\alpha\\-\sin\beta&\cos\beta&0\\\sin\alpha\cos\beta&\sin\alpha\sin\beta&\cos\alpha \end{vmatrix}$
→
Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).
→