The singular solution of the differential equation p3+ px - y = 0 isa)4x3 + 27y2 = 0b)27x3 + 4y2 = 0c)4x2 + 21y2 = 0d)4x3 + 27y3 = 0Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Answer

The given differential equation is:
p^3 * px - y = 0

To find the singular solution, we can use the method of separation of variables.

Step 1: Rewrite the equation in terms of differentials:
p^3 * px - y dx = 0

Step 2: Rearrange the equation to separate the variables:
p^3 * px = y dx

Step 3: Divide both sides by p^3 * x to isolate y and dx:
dy/dx = 1/(p^3 * x)

Step 4: Integrate both sides with respect to x:
∫ dy = ∫ (1/(p^3 * x)) dx

Step 5: Evaluate the integrals:
y = ∫ (1/(p^3 * x)) dx

Step 6: Simplify the integral:
y = (1/p^3) ∫ (1/x) dx

Step 7: Evaluate the integral:
y = (1/p^3) ln|x| + C

Step 8: The general solution of the differential equation is:
y = (1/p^3) ln|x| + C

Step 9: To find the singular solution, we need to eliminate the constant C. For a singular solution, the constant C must be zero.

Step 10: Therefore, the singular solution of the differential equation is:
y = (1/p^3) ln|x|

Step 11: Comparing the given options, we can see that the correct answer is option A:
4x^3 - 27y^2 = 0

Conclusion: The singular solution of the given differential equation is 4x^3 - 27y^2 = 0, which is option A.

Similar Mathematics Doubts

Which of the following statements is incorrect regarding the singular solutions of the differential equations?

The singular solution(s) of the differential equation 8ap3= 27y is/are given by

Consider the differential equation y” + 9y = 0 with the boundary conditions, y(0) = 0, y(2π) = 1, then the differential equation has

The solution of ordinary differential equation dy/dx = y , y(0)=0 is?

Top Courses for Mathematics

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams