
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
163.8k+ views
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.
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.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

GFTI Colleges in India - List, Ranking & Admission Details

Other Pages
NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

JEE Advanced 2025 Notes

Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks
