Question

If $\dfrac{dy}{dx}=y+3>0$ and $y(0)=2$ then $y(ln2)$ is equal to
A. 7
B. 5
C. 13
D.-2

Answer 1

In this question we are given derivative of y with respect to x , so we first integrate both side, then after integration we will get a constant but we are given a condition that $y(0)=2$ so using it will find value of constant, now we know y in terms of x , after that we just need to find $y(ln2)$so by putting $x=ln2$, we will get the answer.

Complete step-by-step answer:
We are given one derivative equation as $\dfrac{dy}{dx}=y+3>0$ its between x and y
After cross multiplying our equation will look like $\dfrac{dy}{y+3}=dx$
So, to find relation between y and x we have to integrate it as $\int{\dfrac{dy}{y+3}}=\int{dx}$
Using integration property $\int{\dfrac{dy}{y}}=\ln (y)$ similarly $\int{\dfrac{dy}{y+3}}=\ln (y+3)$.
After integration expression will look like $\ln (y+3)=x+c$ , where c is a constant
But we have one more equation given $y(0)=2$ , we can put $x=0$ and $y=2$ in equation
$\ln (y+3)=x+c$ to find c
Putting $x=0$ and $y=2$ we get
$\ln (2+3)=0+c$ , and we got $c=\ln (5)$
So finally, our equation will look like $\ln (y+3)=x+\ln (5)$
Now in the question we are asked to find $y(ln2)$
So just put $x=ln2$ in our final equation $\ln (y+3)=x+\ln (5)$
On putting value of x, we get
$\ln (y+3)=\ln (2)+\ln (5)....(1)$
Now we know one property of ln,
$\to \ln (a)+\ln (b)=\ln (ab)$
Using this property, we can write
$\ln (2)+\ln (5)=ln(10).....(2)$
Putting equation (2) in equation (1)
We get $\ln (y+3)=\ln (10)$
Now equate $y+3=10$
Hence, we get $y=7$
Answer is 7

So, the correct answer is “Option A”.

Note: You might have done mistake while writing $\int{\dfrac{dy}{y+3}}=\ln (y+3)$ because we know only basic formula $\int{\dfrac{dy}{y}}=\ln (y)$ so we can to solve $\int{\dfrac{dy}{y+3}}=\ln (y+3)$ we can assume $y+3=t$
Taking derivative we get $dy=dt$ so we can write $\int{\dfrac{dy}{y+3}}=\ln (y+3)$ as $\int{\dfrac{dt}{t}}=\ln (t)$
Now substitute $y+3=t$ we got the result.