Question:
If \(a\) and \(b\) are respectively the order and degree of the differential equation
\[
y^2(y'')^2+3x(y')^{1/3}+x^2y^2=\sin x,
\]
then
If \(a\) and \(b\) are respectively the order and degree of the differential equation
\[
y^2(y'')^2+3x(y')^{1/3}+x^2y^2=\sin x,
\]
then
Show Hint
The order is the highest derivative present. The degree is the power of the highest order derivative after the equation is made polynomial in derivatives.
Updated On: Jun 26, 2026
- \(b=a\)
- \(a=3b\)
- \(b=3a\)
- \(ab=6\)
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The Correct Option is C
Solution and Explanation
Step 1: Find the order.
The given differential equation is \[ y^2(y'')^2+3x(y')^{1/3}+x^2y^2=\sin x. \] The highest order derivative present is \[ y''. \] Therefore, the order is \[ a=2. \]
Step 2: Remove the fractional power of derivative.
The term \[ (y')^{1/3} \] contains a fractional power. To find the degree, the equation must be made free from fractional powers of derivatives.
Rearrange: \[ 3x(y')^{1/3} = \sin x-y^2(y'')^2-x^2y^2. \] Cubing both sides, \[ 27x^3y' = \left(\sin x-y^2(y'')^2-x^2y^2\right)^3. \]
Step 3: Find the degree.
In the cubed equation, the highest order derivative is still \[ y''. \] The highest power of \(y''\) comes from \[ \left(y^2(y'')^2\right)^3. \] So the highest power of \(y''\) is \[ 2\times 3=6. \] Therefore, the degree is \[ b=6. \]
Step 4: Compare \(a\) and \(b\).
We have \[ a=2 \] and \[ b=6. \] Thus, \[ b=3a. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{b=3a} \]
The given differential equation is \[ y^2(y'')^2+3x(y')^{1/3}+x^2y^2=\sin x. \] The highest order derivative present is \[ y''. \] Therefore, the order is \[ a=2. \]
Step 2: Remove the fractional power of derivative.
The term \[ (y')^{1/3} \] contains a fractional power. To find the degree, the equation must be made free from fractional powers of derivatives.
Rearrange: \[ 3x(y')^{1/3} = \sin x-y^2(y'')^2-x^2y^2. \] Cubing both sides, \[ 27x^3y' = \left(\sin x-y^2(y'')^2-x^2y^2\right)^3. \]
Step 3: Find the degree.
In the cubed equation, the highest order derivative is still \[ y''. \] The highest power of \(y''\) comes from \[ \left(y^2(y'')^2\right)^3. \] So the highest power of \(y''\) is \[ 2\times 3=6. \] Therefore, the degree is \[ b=6. \]
Step 4: Compare \(a\) and \(b\).
We have \[ a=2 \] and \[ b=6. \] Thus, \[ b=3a. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{b=3a} \]
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