Solve the following differential equation: dy dx +1=e^ x + y - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x + y}}$

✓

Answer

$\frac{\text{dy}}{\text{dx}}+1 = \text{e}^\text{x+y} .....(1)$
Let $\text{ x}+\text{y} = \text{t}$
$\Rightarrow 1+\frac{\text{dy}}{\text{dx}} = \frac{\text{dt}}{\text{dx}}$
Substituting the value of $\text{x + y = t}$ and $1 + \frac{\text{dy}}{\text{dx}} = \frac{\text{dt}}{\text{dx}} (1),$ we get
$\frac{\text{dt}}{\text{dx}} = \text{e}^1$
$\Rightarrow \text{e}^{-1}\text{dt} = \text{dx}$
$\Rightarrow -\text{e}^{-1} = \text{x}+\text{C}$
$\Rightarrow -\text{e}^{-(\text{x+y})} = \text{x} +\text{C}$ $[\therefore \text{t} = \text{x} + \text{y}]$

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

Evaluate the following integrals:
$\int^\limits{\text{a}}_0\sin{-1}{\sqrt\frac{\text{x}}{\text{a}+\text{x}}}\text{ dx}$

Discuss the applicability of Rolle’s theorem on the function given by.
$\text{f(x)}=\begin{cases}\text{x}^2+1,&\text{if }0\leq\text{x}\leq1\\3-\text{x},&\text{if }1\leq\text{x}\leq2\end{cases}$

Show that the differential equation $\frac{d y}{d x}-\frac{y}{x}+cosec \left(\frac{y}{x}\right)=0$ is homogenous and find the particular solution, given that $y = 0$ when $x = 1$.

If A = { 1, 2, 3}, B = { 4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

If $y^x = e^{y-x}, $Prove that $\frac{\text{dy}}{\text{dx}}=\frac{(1+\log\text{y})^2}{\log\text{y}}$

Evaluate the following integrals:
$\int4\text{x}^3\sqrt{5-\text{x}^2}\text{ dx}$

A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs. 20 per unit for the product A and Rs. 15 per unit for the product B, how many units of each product should be sold to maximize profit?

Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$

Prove that $\text{f}(\text{x})=\sin\text{x}+\sqrt{3}\cos\text{x}$ has maximum value at $\text{x}=\frac{\pi}{6}$.

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 $\vec{\text{b}} \neq \vec{\text{c}}.$