Exercise 9.1 (Revised) - Chapter 9 - Differential Equations - Ncert Solutions class 12 - Maths
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NCERT Class 12 Maths Solutions: Chapter 9 - Differential Equations
Ex 9.1 Question 1.
Determine order and degree (if defined) of differential equations given in Questions 1 to 10 :
$\frac{d^4 y}{d x^4}+\sin \left(y^{\prime \prime \prime}\right)=0$
Answer.
Given: $\frac{d^4 y}{d x^4}+\sin \left(y^{\prime \prime \prime}\right)=0$
The highest order derivative present in the differential equation is $\frac{d^4 y}{d x^4}$ and its order is 4 .
The given differential equation is not a polynomial equation in derivatives as the term $\sin \left(y^{\prime \prime \prime}\right)$ is a T-function of derivative $y^{\prime \prime \prime}$. Therefore the degree is not defined.
Hence, order is 4 and degree is not defined.
Ex 9.1 Question 2.
$y^{\prime}+5 y=0$
Answer.
Given: $y^{\prime}+5 y=0$
The highest order derivative present in the differential equation is $y^{\prime}=\frac{d y}{d x}$ and its order is 1.
The given differential equation is a polynomial equation in derivative $y^{\prime}$ and the highest power raised to highest order derivative $y^{\prime}$ is one, so its degree is 1 .
Hence, order is 1 and degree is 1 .
Ex 9.1 Question 3.
$\left(\frac{d s}{d t}\right)^4+3 s \frac{d^2 s}{d t^2}=0$
Answer.
Given: $\left(\frac{d s}{d t}\right)^4+3 s \frac{d^2 s}{d t^2}=0$
The highest order derivative present in the differential equation is $\frac{d^2 s}{d t^2}$ and its order is 2 .
The given differential equation is a polynomial equation in derivatives and the highest power raised to highest order derivative $\frac{d^2 s}{d t^2}$ is one, so its degree is 1 .
Hence, order is 2 and degree is 1.
Ex 9.1 Question 4.
$\left(\frac{d^2 y}{d x^2}\right)^2+\cos \frac{d y}{d x}=0$
Answer.
Given: $\left(\frac{d^2 y}{d x^2}\right)^2+\cos \frac{d y}{d x}=0$
The highest order derivative present in the differential equation is $\frac{d^2 y}{d x^2}$ and its order is 2 .
The given differential equation is not a polynomial equation in derivatives as the term $\cos \frac{d y}{d x}$ is a T-function of derivative $\frac{d y}{d x}$. Therefore the degree is not defined.
Hence, order is 2 and degree is not defined.
Ex 9.1 Question 5.
$\frac{d^2 y}{d x^2}=\cos 3 x+\sin 3 x$
Answer.
Given: $\frac{d^2 y}{d x^2}=\cos 3 x+\sin 3 x$
The highest order derivative present in the differential equation is $\frac{d^2 y}{d x^2}$ and its order is 2 .
The given differential equation is a polynomial equation in derivatives and the highest power raised to highest order $\frac{d^2 y}{d x^2}=\left(\frac{d^2 y}{d x^2}\right)^1$ is one, so its degree is 1 .
Hence, order is 2 and degree is 1.
Ex 9.1 Question 6.
$\left(y^{\prime \prime \prime}\right)^2+\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4+y^5=0$
Answer.
Given: $\left(y^{\prime \prime \prime}\right)^2+\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4+y^5=0$
The highest order derivative present in the differential equation is $y^{\prime \prime \prime}$ and its order is 3 .
The given differential equation is a polynomial equation in derivatives and the highest power raised to highest order $y^{\prime "}$ is two, so its degree is 2.
Hence, order is 3 and degree is 2 .
Ex 9.1 Question 7.
$y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$
Answer.
Given: $y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0$
The highest order derivative present in the differential equation is $y^{\prime \prime \prime}$ and its order is 3 .
The given differential equation is a polynomial equation in derivatives $y^{\prime \prime \prime}, y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to highest order $y^{\prime "}$ is two, so its degree is 1 .
Hence, order is 3 and degree is 1 .
Ex 9.1 Question 8.
$y^{\prime}+y=e^x$
Answer.
Given: $y^{\prime}+y=e^x$
The highest order derivative present in the differential equation is $y^{\prime}$ and its order is 1 .
The given differential equation is a polynomial equation in derivative $y^{\prime}$. It may be noted that $e^x$ is an exponential function and not a polynomial function but is not an exponential function of derivatives and the highest power raised to highest order derivative $y^{\prime}$ is one so its degree is one.
Hence, order is 1 and degree is 1 .
Ex 9.1 Question 9.
$y^{\prime \prime}+\left(y^{\prime}\right)^2+2 y=0$
Answer.
Given: $y^{\prime \prime}+\left(y^{\prime}\right)^2+2 y=0$
The highest order derivative present in the differential equation is $y^{\prime \prime}$ and its order is 2 .
The given differential equation is a polynomial equation in derivatives $y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to highest order $y^{\prime \prime}$ is one, so its degree is 1 .
Hence, order is 2 and degree is 1 .
Ex 9.1 Question 10.
$y^{\prime \prime}+2 y^{\prime}+\sin y=0$
Answer.
Given: $y^{\prime \prime}+2 y^{\prime}+\sin y=0$
The highest order derivative present in the differential equation is $y^{\prime \prime}$ and its order is 2 .
The given differential equation is a polynomial equation in derivative $y^{\prime \prime}$ and $y^{\prime}$. It may be noted that $\sin y$ is not a polynomial function of $y$, it is a T-function of $y$ but is not a Tfunction of derivatives and the highest power raised to highest order derivative $y^{\prime \prime}$ is one so its degree is one.
Hence, order is 2 and degree is 1.
Ex 9.1 Question 11.
The degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\sin \left(\frac{d y}{d x}\right)+1=0$
(A) 3
(B) 2
(C) 1
(D) Not defined
Answer.
Given: $\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\sin \left(\frac{d y}{d x}\right)+1=0$ $\qquad$
This equation is not a polynomial in derivatives as $\sin \left(\frac{d y}{d x}\right)$ is a T-function of derivative $\frac{d y}{d x}$
Therefore, degree of given equation is not defined.
Hence, option (D) is correct.
Ex 9.1 Question 12.
The order of the differential equation $2 x^2 \frac{d^2 y}{d x^2}-3 \frac{d y}{d x}+y=0$ is:
(A) 2
(B) 1
(C) 0
(D) Not defined
Answer.
Given: $2 x^2 \frac{d^2 y}{d x^2}-3 \frac{d y}{d x}+y=0$
The highest order derivative present in the differential equation is $\frac{d^2 y}{d x^2}$ and its order is 2.
Therefore, option (A) is correct.
