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 "Solve the Following Question.(5 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the area of the region bounded by $\text{y}=\sqrt{\text{x}}$ and $y = x.$

If $\text{A}=\begin{bmatrix}1&0\\-1&7\end{bmatrix},$ find k such that $A^2 - 8A + kI = 0.$

Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

Find the foot of the perpendicular drawn from the point $\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}$ to the line $\vec{\text{r}}=\hat{\text{j}}+2\hat{\text{k}}+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the length of the perpendicylar

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}=\text{at}^2,\text{y}=2\text{at at}\text{ t}=1$

Show that four points whose position vectors are
$6\hat{\text{i}}-7\hat{\text{j}},16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},3\hat{\text{i}}-6\hat{\text{k}},2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

A wholesale dealer deals in two kinds, $A$ and $B$ (say) of mixture of nuts. Each kg of mixture $A$ contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew nuts and 540 grams of hazel nuts. Mixture $A$ costs Rs. 8 per kg. and mixture $B$ costs Rs. 12 per kg. Assuming that mixtures $A$ and $B$ are uniform, use graphical method to determine the number of kg . of each mixture which he should use to minimise the cost of the bag.

In a group of $400$ people, $160$ are smokers and non-vegetarian, $100$ are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are $35\%, 20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?

Solve the following system of equations by matrix method:
$3x + y = 19$
$3x - y = 23$