Solve the following differential equation: dy dx +y= - Vidyadip

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

Similar questions

Evaluate the following integrals:
$\int\text{cosec x}\log({\text{cosec x}-\cot\text{x})}\text{dx}$

A firm manufacturing two types of electric items, $A$ and $B,$ can make a profit of $Rs. 20$ per unit of $A$ and $Rs. 30$ per unit of $B.$ Each unit of $A$ requires $3$ motors and $4$ transformers and each unit of $B$ requires $2$ motors and $4$ transformers. The total supply of these per month is restricted to $210$ motors and $300$ transformers. Type $B$ is an export model requiring a voltage stabilizer which has a supply restricted to $65$ units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

Evaluate the following integrals:
$\int(4\text{x}+1)\sqrt{\text{x}^2-\text{x}-2}\text{dx}$

Find: $\int (2x + 5) \sqrt{10 - 4x - 3x^{2}} \text{ dx}$

Evaluate the following integrals:$\int(\text{x}+1)\text{e}^{\text{x}}\log(\text{xe}^{\text{x}})\text{dx}$

In a bank principal increases at the rate of $5\%$ per year. An amount of $Rs. 1000$ is deposited with this bank, how much will it worth after $10$ years $(e^{0.5_{ =}}1.648).$

If for function $\phi(\text{x})=\lambda\text{x}^2+7\text{x}-4, \phi(5)=97,$ find $\lambda.$

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}$

A publisher sells a hard cover edition of a text book for $Rs. 72.00$ and paperback edition of the same ext for $Rs. 40.00$. Costs to the publisher are $Rs. 56.00$ and $Rs. 28.00$ per book respectively in addition to weekly costs of $Rs. 9600.00$. Both types require $5$ minutes of printing time, although hardcover requires $10$ minutes binding time and the paperback requires only $2$ minutes. Both the printing and binding operations have $4,800$ minutes available each week. How many of each type of book should be produced in order to maximize profit?

If $\text{x}=\cos\theta,\text{y}=\sin^3$ prove that $\text{y}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}^2}\Big)=3\sin^2\theta(5\cos^2\theta-1)$