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To obtain the singular solution of the differential equation in Clairaut’s form it is necessary to calculate
- a)p - discriminant only
- b)c - discriminant only
- c)both the p and c-discriminant
- d)any one of the p and c discriminant
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
To obtain the singular solution of the differential equation in Claira...
If the given differential equation is in Ciairaut's form, then any one of the p and c discriminants can be used for finding the singular solution of the differential equation.
Remark : The equation in Clairaut’s form is
y = xp +f(p) ...(i)
Its general solution is given by
y = cx + f(d) ...(ii)
Differentiating (ii) partially w.r. t. c,
0 = x + f'(c) ...(iii)
The singular solution which is the envelope of (ii) can now be obtained by eliminating c from (ii) and (iii).
Alternately : Differentiate (i) partially w.r. t. p,
0 = x + f'(P) ...(iv)
The singular solution can now be obtained by eliminating p from (i) and (iv).
Remark : The equation in Clairaut’s form is
y = xp +f(p) ...(i)
Its general solution is given by
y = cx + f(d) ...(ii)
Differentiating (ii) partially w.r. t. c,
0 = x + f'(c) ...(iii)
The singular solution which is the envelope of (ii) can now be obtained by eliminating c from (ii) and (iii).
Alternately : Differentiate (i) partially w.r. t. p,
0 = x + f'(P) ...(iv)
The singular solution can now be obtained by eliminating p from (i) and (iv).
Most Upvoted Answer
To obtain the singular solution of the differential equation in Claira...
If the given differential equation is in Ciairaut's form, then any one of the p and c discriminants can be used for finding the singular solution of the differential equation.
Remark : The equation in Clairaut’s form is
y = xp +f(p) ...(i)
Its general solution is given by
y = cx + f(d) ...(ii)
Differentiating (ii) partially w.r. t. c,
0 = x + f'(c) ...(iii)
The singular solution which is the envelope of (ii) can now be obtained by eliminating c from (ii) and (iii).
Alternately : Differentiate (i) partially w.r. t. p,
0 = x + f'(P) ...(iv)
The singular solution can now be obtained by eliminating p from (i) and (iv).
Remark : The equation in Clairaut’s form is
y = xp +f(p) ...(i)
Its general solution is given by
y = cx + f(d) ...(ii)
Differentiating (ii) partially w.r. t. c,
0 = x + f'(c) ...(iii)
The singular solution which is the envelope of (ii) can now be obtained by eliminating c from (ii) and (iii).
Alternately : Differentiate (i) partially w.r. t. p,
0 = x + f'(P) ...(iv)
The singular solution can now be obtained by eliminating p from (i) and (iv).
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Community Answer
To obtain the singular solution of the differential equation in Claira...
To obtain the singular solution of a differential equation in Clairaut's form, follow these steps:
1. Start with the given differential equation in Clairaut's form: y = xy' + f(y')
2. Differentiate both sides of the equation with respect to x.
This yields: y' = y' + xy'' + f'(y')y''
3. Simplify the equation by canceling out the y' terms on both sides.
This gives: 0 = xy'' + f'(y')y''
4. Factor out y'' from the equation.
This results in: 0 = (x + f'(y'))y''
5. Set the factor (x + f'(y')) equal to zero and solve for x.
This gives the value of x where the singular solution occurs.
6. Substitute the value of x obtained in step 5 back into the original equation.
This gives the equation of the singular solution.
Note: The singular solution is a curve that does not depend on any arbitrary constants. It is a special solution that satisfies the differential equation without any additional conditions.
1. Start with the given differential equation in Clairaut's form: y = xy' + f(y')
2. Differentiate both sides of the equation with respect to x.
This yields: y' = y' + xy'' + f'(y')y''
3. Simplify the equation by canceling out the y' terms on both sides.
This gives: 0 = xy'' + f'(y')y''
4. Factor out y'' from the equation.
This results in: 0 = (x + f'(y'))y''
5. Set the factor (x + f'(y')) equal to zero and solve for x.
This gives the value of x where the singular solution occurs.
6. Substitute the value of x obtained in step 5 back into the original equation.
This gives the equation of the singular solution.
Note: The singular solution is a curve that does not depend on any arbitrary constants. It is a special solution that satisfies the differential equation without any additional conditions.
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To obtain the singular solution of the differential equation in Clairaut’s form it is necessary to calculatea)p - discriminant onlyb)c - discriminant onlyc)both the p and c-discriminantd)any one of the p and c discriminantCorrect answer is option 'D'. Can you explain this answer?
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