Solve the following differential equation e^ dy dx =x+1;y(0)=3 - Vidyadip

Question

Solve the following differential equation
$\text{e}^{\frac{\text{dy}}{\text{dx}}}=\text{x}+1;\text{y}(0)=3$

✓

Answer

We have
$\text{e}^{\frac{\text{dy}}{\text{dx}}}=\text{x}+1$
Taking log on both sides, we get
$\frac{\text{dy}}{\text{dx}}\log\text{e}=\log(\text{x}+1)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\log(\text{x}+1) $
$\Rightarrow\text{dy}=\{\log(\text{x}+1)\}\text{dx}$
intregrating both sides ,we get
$\int\text{dy}=\int\{\log(\text{x}+1)\}\text{dx}$
$\Rightarrow\text{y}=\int1\times\log(\text{x}+1)\text{dx}$
$\Rightarrow\text{y}=\log(\text{x}+1)\int1\text{dx}-\int\Big[\frac{\text{d}}{\text{dx}}(\log\text{x}+1)\int1\text{dx}\Big]\text{dx}$
$\Rightarrow\text{y}=\text{x}\log(\text{x}+1)-\int\frac{\text{x}}{\text{x}+1}\ \text{dx}$
$\Rightarrow\text{y}=\text{x}\log(\text{x}+1)-\int\Big(1-\frac{1}{\text{x}+1}\Big)\text{dx}$
$\Rightarrow\text{y}=\text{x}\log(\text{x}+1)-\text{x}+\log(\text{x}+1)+\text{C}\ ...(1)$
It is given that y(0) = 3
$\therefore3=0\times\log(0+1)-0+\log(0+1)+\text{C}$
$\Rightarrow\text{C}=3$
Substituting the value of C in (1), we get
$\text{y}=\text{x}\log(\text{x}+1)+\log(\text{x}+1)-\text{x}+3$
$\Rightarrow\text{y}=(\text{x}+1)\log(\text{x}+1)-\text{x}+3$
Hence, $\text{y}=(\text{x}+1)\log(\text{x}+1)-\text{x}+3$is the solution to the given differential equation.

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

Find the area of the minor segment of the circle x²+ y² = a² cut off by the line $\text{x}=\frac{\text{a}}{2}.$

If the line $\frac{\text{x}-3}{2}=\frac{\text{y}+2}{-1}=\frac{\text{z}+4}{3}$ lies in the plane lx + my - z = 9, then find the value of l² + m².

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

A farmer has a 100 acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man-days for tomatoes and radishes and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs 20 per man-day. Formulate this problem as a LPP to maximize the farmer's total profit.

A factory has three machines X, Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ where $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

Maximum Z = 30x + 20y

$\text{x}+\text{y}\leq8$

$\text{x}+4\text{y}\geq12$

$5\text{x}+8\text{y}=20$

$\text{x},\text{y}\geq0$

A merchant plans to sell two types of personal computers- a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

Find the inverse of the following matrices by using elementry row transformation:

$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\sin\text{x}-\cos\text{x},&\text{if }\text{ x}\neq0\\-1,&\text{if }\text{ x}=0\end{cases}$