ML Aggarwal Class 12 Maths Solutions Section A Chapter 9 Differential Equations Ex 9.1

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ML Aggarwal Class 12 Maths Solutions Section A Chapter 9 Differential Equations Ex 9.1

Very short answer type questions :

Determine the order and the degree (when defined) of each of the following (1 to 20) differential equations:

Question 1.
(i) \(\frac{d y}{d x}\) – cos x = 0 (NCERT)
(ii) \(\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}\) – sin y = 0.
Solution:
Given differential equation be,
\(\frac{d y}{d x}\) – cos x = 0
Which is polynomial in derivatives.
The highest ordered derivative existing in the given diff. eqn. be \(\frac{d y}{d x}\) so its order is 1.
Also, the exponent of \(\frac{d y}{d x}\) is 1.
∴ degree of given differential equation be 1.

(ii) Given differential equation be,
\(\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}\) – sin y = 0
which is polynomial in derivatives.
The higher order derivative existing in given the given diff. eqn. be \(\frac{d y}{d x}\) so its order is 1.
and the highest power of \(\frac{d y}{d x}\) in given diff. eqn. be 2 so its degree is 2.

Question 2.
(i) y’ + y = e^x
(ii) \(\frac{d^2 y}{d x^2}\) = sin 3x + cos 3x. (NCERT)
Solution:
Given differential eqn. be
y’ + y = e^x i.e. \(\frac{d y}{d x}\) + y = e^x
which is polynomial in derivatives.
The highest order derivative present in given differential eqn. be \(\frac{d y}{d x}\) and its exponent be 1.
So order of given diff. eqn. be 1 and its degree be also 1.

(ii) Given diff. eqn. be,
\(\frac{d^2 y}{d x^2}\) = sin 3x + cos 3x
which is polynomial in derivatives.
The highest order derivative existing in given diff. eqn. be \(\frac{d^2 y}{d x^2}\).
So its order be 2.
The highest power of \(\frac{d^2 y}{d x^2}\) be 1.
∴ degree of given differential eqn. be 1.

Question 3.
(i) (x²y – 3x) dy + (x³ – 3y²) dx = 0
(ii) \(\sqrt{1-y^2} d x+\sqrt{1-x^2} d y\) = 0
Solution:
(i) Given differential eqn. can be written as,
\(\frac{d y}{d x}+\frac{x^3-3 y^2}{x^2 y-3 x}\) = 0
which is polynomial in derivatives.
The highest order derivative present in given differential eqn. be \(\frac{d y}{d x}\).
so its order be 1.
Also the degree of given diff. eqn. be the highest exponent of \(\frac{d y}{d x}\) which is 1.
Thus its degree be 1.

(ii) Given differential eqn. can be written as,
\(\frac{d y}{d x}+\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}\) = 0
which is polynomial in derivatives.
The highest order derivatives present in given diff. eqn. be \(\frac{d y}{d x}\) and its order be 1.
The degree of given diff. eqn. be the highest power of \(\frac{d y}{d x}\) which is 1.
So its degree be 1.

Question 4.
(i) \(\frac{1}{x} \cdot \frac{d^2 y}{d x^2}+5 x \frac{d y}{d x}\) = sin 2x
(ii) \(\left(\frac{d y}{d x}\right)^4+3 y \frac{d^2 y}{d x^2}\) = 0
Solution:
(i) Given differential eqn. be,
\(\frac{1}{x} \cdot \frac{d^2 y}{d x^2}+5 x \frac{d y}{d x}\) = sin 2x
which is polynomial in derivatives.
The highest ordered derivative present in given diff. eqn. be \(\frac{d^2 y}{d x^2}\)
So its order is 2.
The degree of given diff. eqn. be the highest exponent of \(\frac{d^2 y}{d x^2}\) which is 1.
So its degree be 1.

Question 5.
(i) (y’)² + y² – 1 = 0
(ii) y” + 5x (y’)² – 6y = log x
Solution:
(i) Given diff. eqn. be,
y’² + y² – 1 = 0
which is polynomial in derivatives.
The highest ordered derivative present in given diff. eqn. be \(\frac{d y}{d x}\) i.e. y’.
so its order be 1.
The degree of the given diff. eqn. is the highest exponent of which is \(\frac{d y}{d x}\).
So its degree be 2.

(ii) Given diff. eqn. be,
y” + 5xy’² – 6y = log x
which is polynomial in derivatives.
The highest ordered derivative in the given diff. eqn. be y”.
so its order be 2.
The degree of the given diff. eqn. be the highest exponent of \(\frac{d^2 y}{d x^2}\) which is 1.
Thus its degree be 1.

Question 6.
(i) x³ (\(\frac{d^2 y}{d x^2}\))² + x (\(\frac{d y}{d x}\))⁴ = 0
(ii) y^(iv) + sin y⁽ⁱ⁾ = 0 (NCERT)
Solution:
(i) Here the highest ordered derivative existing in given differential eqn. be \(\frac{d^2 y}{d x^2}\)
so its order be 2.
Further given differential eqn. can be expressed as polynomial in derivatives.
So the exponent of highest ordered derivative \(\frac{d^2 y}{d x^2}\) which is 2 gives the degree of given differential eqn.

(ii) Here the highest ordered derivative existing in given differential equation be y^(iv) so its order be 4.
Since the given differential eqn. can’t be expressed as polynomial in derivatives as it contains term like sin y⁽¹⁾ which contains infinite number of terms.
So degree of given differential eqn. is not defined.

Question 7.
(i) \(\left(\frac{d^2 y}{d x^2}\right)^3+\frac{d^2 y}{d x^2}+\sin \left(\frac{d y}{d x}\right)\) = 2x
(ii) \(\left(\frac{d^2 y}{d x^2}\right)^3+2 y \frac{d y}{d x}+\sin y=5 x^{\frac{2}{3}}+\log x\)
Solution:
(i) Here, the highest ordered derivative existing in given diff. eqn. be \(\frac{d^2 y}{d x^2}\) and its order be 2.
Clearly given diff. eqn. cannot be expressed as polynomial in \(\frac{d y}{d x}\)
so its degree is not defined [as it contains terms like sin (\(\frac{d y}{d x}\))]

(ii) Clearly the highest ordered derivative existing in given differential eqn. be \(\frac{d^2 y}{d x^2}\)
so its order is 2.
Since each term in derivatives is a polynomial.
So degree of given differential eqn. be highest exponent of \(\frac{d^2 y}{d x^2}\) i.e. 3.

Question 8.
(i) 2x² \(\frac{d^2 y}{d x^2}\) – 3 \(\frac{d y}{d x}\) + y = 0 (NCERT)
(ii) (y”)³ + (y’)² + sin y’ + 1 = 0. (NCERT)
Solution:
(i) Given differential eqn. be
2x² \(\frac{d^2 y}{d x^2}\) – 3 \(\frac{d y}{d x}\) + y = 0,
which is polynomial in derivatives.
The highest ordered derivative present in \(\frac{d^2 y}{d x^2}\) given diff. eqn. be \(\frac{d^2 y}{d x^2}\).
So its order be 2.
The degree of given diff. eqn. is the highest power of \(\frac{d^2 y}{d x^2}\) which is 1.
Thus, its degree be 1.

(ii) Given differential eqn. be
(y”)³ + (y’)² + sin y’ + 1 = 0
The highest order derivative existing in given diff. eqn. be y” and its order be 2.
Thus, the order of given diff. eqn. be 2.
Since the term sin y’ is not polynomial in y’.
∴ The degree of given diff. eqn. is not defined.

Question 9.
(i) x \(\frac{d^2 y}{d x^2}\) = (1 + (\(\frac{d y}{d x}\))²)⁴
(ii) \(\left(\frac{d^4 y}{d x^4}\right)^2=\left(x+\left(\frac{d y}{d x}\right)^2\right)^3\)
Solution:
(i) Clearly the highest ordered derivative existing in given diff. eqn. be \(\frac{d^2 y}{d x^2}\).
so its order be 2.
Clearly the given diff. eqn. can be expressed as polynomial in derivatives.
Hence degree be the highest exponent of \(\frac{d^2 y}{d x^2}\) which is 1.

(ii) Clearly the highest ordered derivative existing in given diff. eqn. be \(\frac{d^4 y}{d x^4}\).
so its order be 4.
Here given differential eqn. can be expressed as polynomial in derivatives.
Hence degree of given diff. eqn. be the highest exponent of \(\frac{d^4 y}{d x^4}\) which is 2.

Question 10.
(i) 3x \(\frac{d y}{d x}\) + \(\frac{5}{\frac{d y}{d x}}\) = y³
(ii) y = x \(\frac{d y}{d x}\) + a \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\)
Solution:
(i) Given differential eqn. be,
3x \(\frac{d y}{d x}\) + \(\frac{5}{\frac{d y}{d x}}\) = y³
⇒ 3x (\(\frac{d y}{d x}\))² + 5 = y³ \(\frac{d y}{d x}\)
which is polynomial in derivatives.
The highest ordered derivative present in given diff. eqn. be \(\frac{d y}{d x}\).
so its order be 1.
The degree of given diff. eqn. be the highest exponent of \(\frac{d y}{d x}\) which is 2.
Thus its degree be 2.

(ii) Given differential eqn. be,
y = x \(\frac{d y}{d x}\) + a \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\)
⇒ y – x \(\frac{d y}{d x}\) = a \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\)
On squaring, we have
(y – x \(\frac{d y}{d x}\))² = a² [1 + (\(\frac{d y}{d x}\))²]
The highest ordered derivative exiosting in given diff. eqn. be \(\frac{d y}{d x}\) and its power 2.
∴ it is of order 1 and degree 2.
Clearly it is a non-linear differential equation.

Question 11.
(i) \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\) = 3x – \(\frac{d y}{d x}\)
(ii) \(5 \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d y}{d x}\right)^2\right)^{\frac{1}{4}}\)
Solution:
(i) Given differential eqn. be,
\(\sqrt{1+\left(\frac{d y}{d x}\right)^2}\) = 3x – \(\frac{d y}{d x}\)
On squaring both sides ;
1 + (\(\frac{d y}{d x}\))² = 9x² + (\(\frac{d y}{d x}\))² – 6x \(\frac{d y}{d x}\)
⇒ 6x \(\frac{d y}{d x}\) = 9x² + 1 = 0
The highest ordered derivative existing in given diff. eqn. be \(\frac{d y}{d x}\) so its order be 1.
Further given duff. eqn. can be expressed as polynomial in derivative.
So degree be the highest exponent of which is 1.

(ii) Given differential eqn. can be written as
\(5 \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d y}{d x}\right)^2\right)^{\frac{1}{4}}\)
Clearly the highest ordered derivative existing in given diff. eqn. be \(\frac{d^2 y}{d x^2}\) and
hence its order be 2.
Here, given diff. eqn. can be expressed as polynomial in derivatives.
So degree of given diff. eqn. be the highest power of which is 4.

Question 11 (old).
\(\left(\frac{d^2 x}{d t^2}\right)^4-7 t\left(\frac{d x}{d t}\right)^3\) = log t.
Solution:
Given differential eqn. be,
\(\left(\frac{d^2 x}{d t^2}\right)^4-7 t\left(\frac{d x}{d t}\right)^3\) = log t
which is polynomial in derivatives.
The highest ordered derivative present in given diff. eqn. be \(\frac{d^2 y}{d x^2}\) and its order be 2.
The degree of given diff. eqn. be the highest exponent of \(\frac{d^2 y}{d x^2}\) which is 4.
Thus its degree be 4.

Question 12.
(i) y” + (y’)² + 2y = 0 (NCERT)
(ii) y” + 2y’ + sin y = 0
Solution:
(i) Given differential eqn. be,
y” + (y’)² + 2y = 0
which is polynomial in derivatives.
The highest order derivative present in given diff. eqn. be y” and its order be 2.
The degree of given differential equation is the highest exponent of y” which is 1.
Thus its degree be 1.

(ii) Given differential eqn. be,
y” + 2y’ + sin y = 0
which is polynomial in derivatives.
The highest order derivative present in given diff. eqn. be y” and its order is 2.
The degree of given diff. eqn. be the highest exponent of y” which is 1.
Thus its degree be 1.

Question 12 (old).
y^(iv) + sin y”’ = 0 (NCERT)
Solution:
Given diff. eqn. be,
y^(iv) + sin y”’ = 0
The order of given diff. eqn. be the highest order derivative present in given diff. eqn. which is y^(iv) and its order be 4.
Here the term sin y”’ is not polynomial in y”’.
Thus, the degree of given differential eqn. is not defined.

Question 13.
(i) (y”’)² + (y”)³ + (y’)⁴ + y⁵ = 0 (NCERT)
(ii) (1 – (y’)²)^3/2 = ky”
Solution:
(i) Given diff. eqn. be,
(y”’)² + (y”)³ + (y’)⁴ + y⁵ = 0
which is polynomial in derivatives.
The highest order derivative present in given diff. eqn. be y”’ so its order be 3.
The degree of given diff. eqn. is the highest exponent of y”’, which is 2.
Thus degree of given diff. eqn. be 2.

(ii) Given diff. eqn. can be written as
(1 – (y’)²)³ = (ky”)²
Clearly the highest ordered derivative existing in given duff. eqn. be y” so its order 2.
Here the given duff. eqn. can be expressed as polynomial in derivatives.
Thus its degree be the highest exponent of y” which is 2.

Question 14.
Write the sum of the order and the degree of the differential equation \(\left(\frac{d y}{d x}\right)^5+3 x y\left(\frac{d^3 y}{d x^3}\right)^2+y\left(\frac{d^2 y}{d x^2}\right)^4\) = 0.
Solution:
Clearly the highest ordered derivative existing in given diff. eqn. be \(\frac{d^3 y}{d x^3}\).
so its order be 3.
Clearly the given duff. eqn. can be expressed as polynomial in derivatives.
So degree of given diff. eqn. be the highest exponent of \(\frac{d^3 y}{d x^3}\) which is 2.
∴ required sum = 3 + 2 = 5.

Question 15.
Find the product of the order and degree of the following differential equation :
x (\(\frac{d^2 y}{d x^2}\))² + (\(\frac{d y}{d x}\))² + y² = 0.
Solution:
The given differential eqn. be
x (\(\frac{d^2 y}{d x^2}\))² + (\(\frac{d y}{d x}\))² + y² = 0
which is polynomial in derivatives.
The highest order derivative present in given diff. eqn. be \(\frac{d^2 y}{d x^2}\).
So its order be 2.
The degree of given diff. eqn. be the highest exponent of \(\frac{d^2 y}{d x^2}\) which is 2.
Thus, its degree be 2.
∴ required product of order and degree of given differential equation 2 × 2 = 4.

Question 20 (old).
y = px + \(\sqrt{a^2 p^2+b^2}\), where p = \(\frac{d y}{d x}\).
Solution:
Given diff. eqn. can be written as
(y – px)² = a²p² + b²; where p = \(\frac{d y}{d x}\)
Clearly it is a polynomial in p.
The highest order derivative existing in diff. eqn. be p i.e. \(\frac{d y}{d x}\) and its power 2.
Thus given diff. eqn. is of order 1 and degree 2.
Clearly it is a non-linear differential equation.

Question 21 (old).
Write the sum of the order and the degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^2-\left(\frac{d y}{d x}\right)^3\) = y³.
Solution:
Given differential equation be,
\(\left(\frac{d^2 y}{d x^2}\right)^2-\left(\frac{d y}{d x}\right)^3\) = y³
which is polynomial in derivatives.
The highest order derivative present in given
diff. eqn. be \(\frac{d^2 y}{d x^2}\).
so its order be 2.
The degree of given diff. eqn. be the highest power of \(\frac{d^2 y}{d x^2}\) which is 2.
So its degree be 2 and required sum = 2 + 2 = 4.