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The general solution of the second order
differrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is
(a) sqrt(x) * (c_{1} + c_{2} * log(x))
(b) sqrt(x) * (c_{1} + c_{2}*x)
(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)
(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))?
differrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is
(a) sqrt(x) * (c_{1} + c_{2} * log(x))
(b) sqrt(x) * (c_{1} + c_{2}*x)
(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)
(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))?
Most Upvoted Answer
The general solution of the second orderdifferrential equation 4x ^ 2 ...
Understanding the Differential Equation
The given second-order differential equation is:
4x^2 * d^2y/dx^2 + y = 0.
This is a linear homogeneous equation with variable coefficients. To solve it, we can look for solutions of the form y = x^r, where r is a constant.
Substituting the Form
1. Substitute y = x^r into the equation.
2. Calculate the second derivative d^2y/dx^2.
3. Replace y and its derivatives in the original equation.
Finding the Characteristic Equation
By substituting, we obtain a characteristic equation in terms of r, which is typically a polynomial in r. Solving this polynomial will yield the characteristic roots.
General Solution Structure
For a second-order differential equation, the general solution typically takes the form:
y(x) = c1 * y1(x) + c2 * y2(x),
where y1 and y2 are linearly independent solutions derived from the roots of the characteristic equation.
Identifying the Correct Option
After solving, the solutions will involve terms that can be related to square roots and logarithmic functions. The general observation leads us to the following options:
- (a) sqrt(x) * (c1 + c2 * log(x))
- (b) sqrt(x) * (c1 + c2 * x)
- (c) sqrt(x) * (c1 + c2 * e^x)
- (d) c1 * sqrt(x) + c2 * 1/(sqrt(x))
Through analysis, option (a) is the most suitable as it incorporates both the square root term and the logarithmic function, aligning with the expected behavior of solutions to such differential equations.
Conclusion
Thus, the general solution of the differential equation is:
Answer: (a) sqrt(x) * (c1 + c2 * log(x))
The given second-order differential equation is:
4x^2 * d^2y/dx^2 + y = 0.
This is a linear homogeneous equation with variable coefficients. To solve it, we can look for solutions of the form y = x^r, where r is a constant.
Substituting the Form
1. Substitute y = x^r into the equation.
2. Calculate the second derivative d^2y/dx^2.
3. Replace y and its derivatives in the original equation.
Finding the Characteristic Equation
By substituting, we obtain a characteristic equation in terms of r, which is typically a polynomial in r. Solving this polynomial will yield the characteristic roots.
General Solution Structure
For a second-order differential equation, the general solution typically takes the form:
y(x) = c1 * y1(x) + c2 * y2(x),
where y1 and y2 are linearly independent solutions derived from the roots of the characteristic equation.
Identifying the Correct Option
After solving, the solutions will involve terms that can be related to square roots and logarithmic functions. The general observation leads us to the following options:
- (a) sqrt(x) * (c1 + c2 * log(x))
- (b) sqrt(x) * (c1 + c2 * x)
- (c) sqrt(x) * (c1 + c2 * e^x)
- (d) c1 * sqrt(x) + c2 * 1/(sqrt(x))
Through analysis, option (a) is the most suitable as it incorporates both the square root term and the logarithmic function, aligning with the expected behavior of solutions to such differential equations.
Conclusion
Thus, the general solution of the differential equation is:
Answer: (a) sqrt(x) * (c1 + c2 * log(x))
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The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))?
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The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))?.
The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The general solution of the second orderdifferrential equation 4x ^ 2 * dy^2/(dx ^ 2) + y = 0 is(a) sqrt(x) * (c_{1} + c_{2} * log(x))(b) sqrt(x) * (c_{1} + c_{2}*x)(c) sqrt(x) * (c_{1} + c_{2} * e ^ x)(d) c_{1} * sqrt(x) + c_{2} * 1/(sqrt(x))?.
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