KCET · Maths · Differential Equations
If \(m\) and \(n\) are order and degree of the differential equation
\(\left(y^{\prime \prime}\right)^{5}+4 \cdot \frac{\left(y^{\prime \prime}\right)^{3}}{y^{\prime \prime \prime}}+y^{\prime \prime \prime}=\sin x\), then
- A \(m=3, n=5\)
- B \(m=3, n=1\)
- C \(m=3, n=3\)
- D \(m=3, n=2\)
Answer & Solution
Correct Answer
(D) \(m=3, n=2\)
Step-by-step Solution
Detailed explanation
Given differential equation is,
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{5}+4 \cdot \frac{\left(\frac{d^{2} y}{d x^{2}}\right)^{3}}{\left(\frac{d^{3} y}{d x^{3}}\right)}+\left(\frac{d^{3} y}{d x^{3}}\right)=\sin x\)
\(\Rightarrow\left(\frac{d^{2} y}{d x^{2}}\right)^{5} \cdot\left(\frac{d^{3} y}{d x^{3}}\right)+4 \cdot\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d^{3} y}{d x^{3}}\right)^{2}\)
\(=\sin x \cdot\left(\frac{d^{3} y}{d x^{3}}\right)\)
Now, \(m=\) order \(=\) highest order derivative \(=3\)
and \(n=\) degree \(=\) power of highest order derivative \(=2\)
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{5}+4 \cdot \frac{\left(\frac{d^{2} y}{d x^{2}}\right)^{3}}{\left(\frac{d^{3} y}{d x^{3}}\right)}+\left(\frac{d^{3} y}{d x^{3}}\right)=\sin x\)
\(\Rightarrow\left(\frac{d^{2} y}{d x^{2}}\right)^{5} \cdot\left(\frac{d^{3} y}{d x^{3}}\right)+4 \cdot\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d^{3} y}{d x^{3}}\right)^{2}\)
\(=\sin x \cdot\left(\frac{d^{3} y}{d x^{3}}\right)\)
Now, \(m=\) order \(=\) highest order derivative \(=3\)
and \(n=\) degree \(=\) power of highest order derivative \(=2\)
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