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

Hint: 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.