Continuous practice using ISC Class 12 OP Malhotra Solutions Chapter 17 Differential Equations Ex 17(a) can lead to a stronger grasp of mathematical concepts.
State the order and the degree of the following differential equations :
Question 1.
\(\frac { dy }{ dx }\) = sin x
Solution:
Given differential eqn. be,
\(\frac { dy }{ dx }\) = sin x
Here, order of highest ordered derivative existing in diff. eqn. be 1 and its power be 1,
∴ order of differential eqn. be 1 and degree of differential eqn. be 1.
Question 2.
x²(\(\frac { dy }{ dx }\))² + 2y²x = 0
Solution:
Given diff. eqn. be,
x²(\(\frac { dy }{ dx }\))² + 2y²x = 0
Here the order of highest ordered derivative existing in diff. eqn. be 1. The exponent of highest ordered derivative be 2.
∴ Order and degree of diff. eqn. be 1 and 2.
Question 3.
\(\frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2\) + x = 0
Solution:
Given diff. eqn. be,
\(\frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2\) + x = 0
The highest ordered derivative existing in differential eqn. be \(\frac{d^2 y}{d x^2}\) and its order be 2.
∴ order of given diff. eqn. be 2. The exponent of highest ordered derivative be 1.
∴ degree of given diff. eqn. be 1.
Question 4.
\(\left(\frac{d^2 y}{d x^2}\right)^2+\frac{d y}{d x}\) – xy = 0
Solution:
Given differential eqn. be
\(\left(\frac{d^2 y}{d x^2}\right)^2+\frac{d y}{d x}\) – xy = 0
Here, the order of the highest ordered derivative existing in the differential eqn. be 2 and its power 2.
Thus, order of given diff. eqn. be 2 and degree of given differential eqn. be 2.
Question 5.
\(\frac{d^3 y}{d x^3}-5 \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^4\) – 5x = 0
Solution:
Given differential eqn. be,
\(\frac{d^3 y}{d x^3}-5 \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^4\) – 5x = 0
Here the highest ordered derivative existing in given diff., eqn. be \(\frac{d^3 y}{d x^3}\) and its order be 3. Thus order of given diff. eqn. be 3. The power of \(\frac{d^3 y}{d x^3}\) in given diff. eqn. be 1. Thus degree of given diff. eqn. be 1.
Question 6.
y = \(x \frac{d y}{d x}+\frac{a}{\frac{d y}{d x}}\)
Solution:
Given differential eqn. can be written as \(x \frac{d y}{d x}+\frac{a}{\frac{d y}{d x}}\)
y\(\frac { dy }{ dx }\) = x(\(\frac { dy }{ dx }\))² + a
Here the order of highest ordered derivative existing in diff. eqn. be 1 and its power be 2.
Thus order of given diff. eqn. be 1.
and the degree of given diff. eqn. be 2.
Question 7.
\((\sqrt{a+x})\left(\frac{d y}{d x}\right)\) + x = 0
Solution:
The given diff. eqn. be,
\(\sqrt{a+x} \frac{d y}{d x}\) + x = 0
Here, the order of highest ordered derivative existing in diff. eqn. be 1 and its power be also equal to 1.
Thus, order of given diff. eqn. be 1 and its degree be also equal to 1.
Question 8.
\(x \sqrt{1-y^2} d x+y \sqrt{1-x^2} d y\) = 0
Solution:
Given differential eqn. can be written as :
\(x \sqrt{1-y^2} d x+y \sqrt{1-x^2} d y\) = 0
Here, the order of highest ordered derivative existing in given diff. eqn. be 1 and its power be also 1.
Thus order of given differential eqn. be 1 and degree of given diff. eqn. be 1.
Question 9.
\(\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=5 \frac{d^2 y}{d x^2}\)
Solution:
Given diff. eqn. be,
\(\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=5 \frac{d^2 y}{d x^2}\);
On squaring both sides ; we have
\(\left[1+\left(\frac{d y}{d x}\right)^2\right]^3=25\left(\frac{d^2 y}{d x^2}\right)^2\)
Here the order of highest ordered derivative existing in given diff. eqn. be 2 and its power be equal to 2. Hence the order and degree of given diff. eqn. be 2 and 2.
Question 10.
y = \(x \frac{d y}{d x}+a \sqrt{1+\left(\frac{d y}{d x}\right)^2}\)
Solution:
Given diff. eqn. be written as,
y – x\(\frac{d y}{d x}=a \sqrt{1+\left(\frac{d y}{d x}\right)^2}\);
On squaring ; we have
\(\left(y-x \frac{d y}{d x}\right)^2=a^2\left[1+\left(\frac{d y}{d x}\right)^2\right]\)
Here the order of highest ordered derivative existing in diff. eqn. be 1 and its power be 2.
∴ order of given diff. eqn. be 1. and the degree of given diff. eqn. be 2.
Question 11.
\(\left(\frac{d^2 y}{d x^2}\right)^2=\left(\frac{d y}{d x}\right)^2\)
Solution:
Given diff. eqn. can be written as,
\(\left(\frac{d^2 y}{d x^2}\right)^2=\left(\frac{d y}{d x}\right)^2\)
Here, the order of the highest ordered derivative existing in the diff. eqn. be 2 and its power be 3.
Thus the order of given diff. eqn. be 2 and degree of given diff. eqn. be 3.
Question 12.
\(\frac{d y}{d x}=\frac{x}{d y / d x}\)
Solution:
Given differential eqn. can be written as
(\(\frac { dy }{ dx }\))² = x
Here the order of highest ordered derivative existing in given diff. eqn. be 1 and its power be 2.
Thus, the order of given diff. eqn. be 1 and degree of given differential eqn. be 2.