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Can you explain the answer of this question below:
The differential equation (d2y/dx2)2/3 = (y + (dy/dx))1/2 is of
- A:second order, fourth degree
- B:first order, fourth degree
- C:second order, third degree
- D:second order, second degree
- E:undefined
The answer is a.
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Can you explain the answer of this question below:The differential equ...
Explanation:
The given differential equation is (d^2y/dx^2)^(2/3) = (y(dy/dx))^(1/2)
Let's simplify the given differential equation by assuming y = y(x).
1. Take the derivative of both sides w.r.t x.
(d^2y/dx^2)^(2/3) = (y(dy/dx))^(1/2)
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(dy/dx))^(-1/2) * (d/dx)(y(dy/dx))
2. Simplify the above expression by substituting (dy/dx) = p.
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(p))^(-1/2) * (dy/dx)(y(p))
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(p))^(-1/2) * p(y(p))
3. Take the derivative of both sides w.r.t x.
(2/3)(-1/3)(d^2y/dx^2)^(-4/3) * (d^3y/dx^3)^2 - (2/3)(d^2y/dx^2)^(-1/3) * (d^4y/dx^4) = (1/2)(-1/2)(y(p))^(-3/2) * (dy/dx)^2(y(p)) + (1/2)(y(p))^(-1/2) * (d^2y/dx^2)(dy/dx)
(2/9)(d^2y/dx^2)^(-4/3) * (d^3y/dx^3)^2 - (2/3)(d^2y/dx^2)^(-1/3) * (d^4y/dx^4) = -(1/4)(y(p))^(-3/2) * p^2(y(p)) + (1/2)(y(p))^(-1/2) * (d^2y/dx^2)p
4. Substitute (d^2y/dx^2) = q and simplify.
(2/9)q^(-4/3) * (d^3y/dx^3)^2 - (2/3)q^(-1/3) * (d^2q/dx^2) = -(1/4)(y(p))^(-3/2) * p^2(y(p)) + (1/2)(y(p))^(-1/2) * q(p)
5. The above equation is a second-order differential equation with fourth-degree. Therefore, the correct option is (a) second order, fourth degree.
The given differential equation is (d^2y/dx^2)^(2/3) = (y(dy/dx))^(1/2)
Let's simplify the given differential equation by assuming y = y(x).
1. Take the derivative of both sides w.r.t x.
(d^2y/dx^2)^(2/3) = (y(dy/dx))^(1/2)
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(dy/dx))^(-1/2) * (d/dx)(y(dy/dx))
2. Simplify the above expression by substituting (dy/dx) = p.
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(p))^(-1/2) * (dy/dx)(y(p))
(2/3)(d^2y/dx^2)^(-1/3) * d^3y/dx^3 = (1/2)(y(p))^(-1/2) * p(y(p))
3. Take the derivative of both sides w.r.t x.
(2/3)(-1/3)(d^2y/dx^2)^(-4/3) * (d^3y/dx^3)^2 - (2/3)(d^2y/dx^2)^(-1/3) * (d^4y/dx^4) = (1/2)(-1/2)(y(p))^(-3/2) * (dy/dx)^2(y(p)) + (1/2)(y(p))^(-1/2) * (d^2y/dx^2)(dy/dx)
(2/9)(d^2y/dx^2)^(-4/3) * (d^3y/dx^3)^2 - (2/3)(d^2y/dx^2)^(-1/3) * (d^4y/dx^4) = -(1/4)(y(p))^(-3/2) * p^2(y(p)) + (1/2)(y(p))^(-1/2) * (d^2y/dx^2)p
4. Substitute (d^2y/dx^2) = q and simplify.
(2/9)q^(-4/3) * (d^3y/dx^3)^2 - (2/3)q^(-1/3) * (d^2q/dx^2) = -(1/4)(y(p))^(-3/2) * p^2(y(p)) + (1/2)(y(p))^(-1/2) * q(p)
5. The above equation is a second-order differential equation with fourth-degree. Therefore, the correct option is (a) second order, fourth degree.
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Can you explain the answer of this question below:The differential equation (d2y/dx2)2/3 = (y + (dy/dx))1/2 is ofA:second order, fourth degreeB:first order, fourth degreeC:second order, third degreeD:second order, second degreeE:undefinedThe answer is a.
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