Question

If \[\dfrac{{dy}}{{dx}} = {e^{ - 2y}}\] and \[y = 0\] for \[x = 5\]. Then find the value of \[x\] for \[y = 3\].
A. \[{e^5}\]
B. \[{e^6} + 1\]
C. \[\dfrac{{{e^9} + 1}}{2}\]
D. \[\log_{e}6 \]

Answer 1

Hint: Here, the first order differential equation is given. First, simplify the given equation by rearranging the terms. Then, integrate both sides of the equation with respect to the corresponding variables. After that, solve the integrals by using the U-substitution method on the left-hand side and get the solution of the differential equation. Then, substitute the given values in the solution and find the value of the integration constant. In the end, substitute the values and integration constant in the solution and solve it to get the required answer.

Formula Used: \[\int {ndx} = x + c\] , where \[n\] is a number and \[c\] is an integration constant.
\[\int {{e^x}dx} = {e^x} + c\]

Complete step by step solution: The given differential equation is \[\dfrac{{dy}}{{dx}} = {e^{ - 2y}}\] and \[y = 0\] for \[x = 5\].

Simplify the given equation.
\[\dfrac{{dy}}{{{e^{ - 2y}}}} = dx\]
\[ \Rightarrow {e^{2y}}dy = dx\]
Now integrate both sides with respect to the corresponding variables.
\[\int {{e^{2y}}dy} = \int {dx} \] \[.....\left( 1 \right)\]
It is difficult to find the integral of the above equation.
So, apply the substitution method on the left-hand side.
Substitute \[2y = u\].
Differentiating the substitute equation, we get
\[2dy = du\]
\[ \Rightarrow dy = \dfrac{{du}}{2}\]
Then, we get the equation \[\left( 1 \right)\] as
\[\int {\dfrac{{{e^u}}}{2}du} = \int {dx} \]
Take the constant term outside of the integral.
\[\dfrac{1}{2}\int {{e^u}du} = \int {dx} \]
Now solve the integrals by using the formulas \[\int {{e^x}dx} = {e^x} + c\], and \[\int {ndx} = x + c\].
We get,
\[\dfrac{1}{2}{e^u} = x + c\]
Resubstitute the value of \[u\].
\[\dfrac{1}{2}{e^{2y}} = x + c\]
The solution of the given differential equation is \[\dfrac{1}{2}{e^{2y}} = x + c\].

Now to find the value of the integration constant \[c\], substitute \[y = 0\] and \[x = 5\] in the solution.
\[\dfrac{1}{2}{e^{2\left( 0 \right)}} = 5 + c\]
\[ \Rightarrow \dfrac{1}{2}{e^0} = 5 + c\]
\[ \Rightarrow \dfrac{1}{2}\left( 1 \right) = 5 + c\]
\[ \Rightarrow c = \dfrac{1}{2} - 5\]
\[ \Rightarrow c = - \dfrac{9}{2}\]
We have to find the value of \[x\] for \[y = 3\].
So, substitute \[y = 3\], and \[c = - \dfrac{9}{2}\] in the equation \[\left( 1 \right)\].
Then,
 \[\dfrac{1}{2}{e^{2\left( 3 \right)}} = x - \dfrac{9}{2}\]
Solve the above equation.
\[\dfrac{1}{2}{e^6} = \dfrac{{2x - 9}}{2}\]
\[ \Rightarrow {e^6} = 2x - 9\]
\[ \Rightarrow {e^6} + 9 = 2x\]
Divide both sides by 2.
\[x = \dfrac{{{e^6} + 9}}{2}\]

Option ‘C’ is correct

Note: Students often forget to calculate the value of the integration constant while solving this type of questions.
Always remember to calculate the value of the integration constant by using the given values or information.