The solution of the equation d y d x =e^ x+y +e^y x^2 is : - Vidyadip

MCQ

The solution of the equation $\frac{d y}{d x}=e^{x+y}+e^y x^2$ is :

A
$e^x+e^y=\frac{x^3}{3}+c$
B
$e^{-x}+e^y+\frac{x^3}{3}=c$
C
$e^{-x}+e^{-y}-\frac{x^3}{3}=c$
✓
$e^x+e^{-y}+\frac{x^3}{3}=c$

✓

Answer

Correct option: D.

$e^x+e^{-y}+\frac{x^3}{3}=c$

(D)
$
\frac{d y}{d x}=e^{x+y}+e^y x^2
$
$
\begin{array}{rlrl}
\frac{d y}{d x} & =e^x \times e^y+e^y x^2 \\
\frac{d y}{d x} & =e^y\left(e^x+x^2\right) \\
\Rightarrow & \frac{d y}{e^y} & =\left(e^x+x^2\right) d x \\
\Rightarrow & e^{-y} d y & =\left(e^x+x^2\right) d x \\
\text { Hence } & \int e^{-y} d y & =\int\left(e^x+x^2\right) d x \\
& -e^{-y} & =e^x+\frac{x^3}{3}+c \\
\text { or } & -e^x+e^{-y}+\frac{x^3}{3} & =c
\end{array}
$
Therefore the correct choice is (D).

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 "M.C.Q (1 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

${d \over {dx}}\left( {{1 \over {{x^4}\sec x}}} \right) = $

The value of $ \cos \left( \sin^{-1} \left( \frac {2}{3} \right) \right)$ is equal to :

$ \frac {\sqrt4}{8}$
$ \frac {\sqrt4}{3}$
$ \frac {\sqrt5}{4}$
$ \frac {\sqrt5}{3}$

Two dice are thrown simultaneously. The probability of getting a pair of aces is

$\frac{1}{36}$
$\frac{1}{3}$
$\frac{1}{6}$
None of these.

If the shortest distance between the lines

$ \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(2+\lambda) \hat{\mathrm{i}}+(1-3 \lambda) \hat{\mathrm{j}}+(3+4 \lambda) \hat{\mathrm{k}}, \lambda \in \mathbb{R} $

$ \mathrm{L}_2: \overrightarrow{\mathrm{r}}=2(1+\mu) \hat{\mathrm{i}}+3(1+\mu) \hat{\mathrm{j}}+(5+\mu) \hat{k}, \mu \in \mathbb{R}$

is $\frac{\mathrm{m}}{\sqrt{\mathrm{n}}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then the value of $\mathrm{m}+\mathrm{n}$ equals.

In a study about a pandemic, data of $900$ persons was collected. It was found that

$190$ persons had symptom of fever,

$220$ persons had symptom of cough,

$220$ persons had symptom of breathing problem,

$330$ persons had symptom of fever or cough or both,

$350$ persons had symptom of cough or breathing problem or both,

$340$ persons had symptom of fever or breathing problem or both,

$30$ persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is. . . . .

If $\sin ({\cot ^{ - 1}}(x + 1) = \cos ({\tan ^{ - 1}}x)$, then $ x =$

Function f(x) = a^x is increasing on R, if:

a > 0
a < 0
0 < a < 1
a > 1

The shortest distance between the lines $\frac{{x - 3}}{2} = \frac{{y + 15}}{{ - 7}} = \frac{{z - 9}}{5}$ and $\frac{{x + 1}}{2} = \frac{{y - 1}}{1} = \frac{{z - 9}}{{ - 3}}$ is

The order and degree of the differential equation ${\left[ {4 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^{2/3}} = \frac{{{d^2}y}}{{d{x^2}}}$ are

Let $A$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is ..................