Mathematics Exam  >  Mathematics Questions  >  Let y1(x) = 1 + x and y2(x) = ex be two solut... Start Learning for Free
Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is . 
  • a)
     y(1)= 2, y’(1) = 1 
  • b)
     y(0) =1, y’(0) = 2 
  • c)
     y(0)= 2, y’(0) = 1 
  • d)
     None of these 
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x...
Since y1 = 1 + x and y2 = ex be two solutions of 
y’’(x) + P(x) y’(x) + Q(x) y(x) = 0 ............(1) 
then will satisfy (1) 
so if y1 = 1 + x if y2 = ex
and P(x) + Q(1 + x) = 0 
(1 + P + Q)ex = 0 
⇒ P + Q(1 + x) = 0 
P + Q + 1 = 0 
on solving them we get 
on solving equation set (1) 
we get no solution 
Thus for the conditions y(0) = 2, y’(0) = 1 
differential equation has no solution
Free Test
Community Answer
Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x...
Explanation:

Given Information:
Two solutions of the differential equation are y1(x) = 1 + x and y2(x) = ex.

Conditions for No Solution:
For the given differential equation to have no solution, the two solutions must be linearly dependent. This means that one solution can be expressed as a constant multiple of the other solution.

Checking Linear Dependence:
Let's check if y1(x) = 1 + x and y2(x) = ex are linearly dependent:
y2(x) = cy1(x), where c is a constant
ex = c(1 + x)
ex = c + cx

Solving for c:
Comparing the coefficients of x on both sides, we get:
c = 1 and c = 0, which is a contradiction

Conclusion:
Since the solutions y1(x) = 1 + x and y2(x) = ex are not linearly dependent, the differential equation will have a solution for any set of initial conditions.

Correct Option:
The correct option is (c) y(0) = 2, y'(0) = 1, as all initial conditions will yield a solution for the given differential equation.
Explore Courses for Mathematics exam
Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer?
Question Description
Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice Let y1(x) = 1 + x and y2(x) = ex be two solutions of y”(x) + p(x)y’(x) + Q(x)y(x) = 0 then the set of initial conditions for which the above differential equation has No solution is .a)y(1)= 2, y’(1) = 1b)y(0) =1, y’(0) = 2c)y(0)= 2, y’(0) = 1d)None of theseCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Mathematics tests.
Explore Courses for Mathematics exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev