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The ex 9.1 class 12 maths NCERT solutions provide a clear and structured approach to understanding the basics of differential equations. This part of the class 12 maths exercise 9.1 introduces students to fundamental concepts like order, degree, and formation of differential equations. With step-by-step solutions, the class 12 maths NCERT solutions chapter 9 exercise 9.1 help learners strengthen their foundation in this topic. Whether you’re revising for exams or practicing key problems, the exercise 9.1 class 12 maths NCERT solutions serve as a reliable resource. For students aiming to excel, the chapter 9 class 12 maths NCERT solutions on differential equations class 12 NCERT solutions are an indispensable guide.
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EXERCISE 9.1
1. Determine order and degree (if defined) of differential equations given
\(\frac{d^{4} y}{d x^{4}}+\sin \left(y^{\prime \prime \prime}\right)=0\)
\(\frac{d^{4} y}{d x^{4}}+\sin \left(y^{\prime \prime \prime}\right)=0\)
Answer
It is given that equation is \( \frac{d^{4} y}{d x^{4}}+\sin \left(y^{\prime \prime \prime}\right)=0 \)
\(\Rightarrow y^{\prime \prime\prime \prime}+\sin \left(y^{\prime \prime \prime}\right)=0\)
We can see that the highest order derivative present in the differential is \(y^{\prime \prime \prime\prime}\).
Thus, its order is four. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
2. Determine order and degree (if defined) of differential equations given
\(y^{\prime}+5 y=0\)
\(y^{\prime}+5 y=0\)
Answer
It is given that equation is \( y^{\prime}+5 y=0 \)
We can see that the highest order derivative present in the differential is \( y^{\prime} \).
Thus, its order is one. It is polynomial equation in \(y^{\prime}\). The highest power raised to \( y^{\prime} \) is \(1 \).
Therefore, its degree is one.
3. Determine order and degree (if defined) of differential equations given
\(\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0\)
\(\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0\)
Answer
It is given that equation is \( \left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0 \)
We can see that the highest order derivative present in the given differential equation is \( \frac{d^{2} s}{d t^{2}} \)
Thus, its order is two. It is polynomial equation in \( \frac{d^{2} s}{d t^{2}} \) and \( \frac{d s}{d t} \)
Therefore, its degree is one.
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4. Determine order and degree (if defined) of differential equations given
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0\)
\(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0\)
Answer
It is given that equation is \( \left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0 \)
We can see that the highest order derivative present in the given differential equation is \( \frac{d^{2} y}{d x^{2}} \).
Thus, its order is two. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
5. Determine order and degree (if defined) of differential equations given
\(\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x\)
\(\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x\)
Answer
It is given that equation is \( \frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x \)
\(\Rightarrow \frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x=0\)
We can see that the highest order derivative present in the given differential equation is
Thus, its order is two. It is polynomial equation in \( \frac{d^{2} s}{d t^{2}} \) and the power is 1.
Therefore, its degree is one.
6. Determine order and degree (if defined) of differential equations given
\(\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0\)
\(\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0\)
Answer
It is given that equation is \( \left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0 \)
We can see that the highest order derivative present in the differential is \(y^{\prime \prime \prime}\).
Thus, its order is three. It is polynomial equation in \( y^{\prime \prime \prime}, y^{\prime \prime} \) and \( y^{\prime} \)
So, the highest power raised to \( y^{\prime \prime \prime} \) is \(2 \).
Therefore, its degree is two.
7. Determine order and degree (if defined) of differential equations given
\(y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0\)
\(y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0\)
Answer
It is given that equation is \( y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0 \)
We can see that the highest order derivative present in the differential is \( y^{\prime \prime \prime} \).
Thus, its order is three. It is polynomial equation in \( y^{\prime \prime \prime} , y^{\prime \prime} \) and \( y ^{\prime}\)
So, the highest power raised to \( y^{\prime \prime} \) is \(1 \).
Therefore, its degree is one.
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8. Determine order and degree (if defined) of differential equations given
\(y^{\prime}+y=e^{x}\)
\(y^{\prime}+y=e^{x}\)
Answer
It is given that equation is \( y^{\prime}+y=e^{x} \)
\(\Rightarrow y^{\prime}+y-\mathrm{e}^{x}=0\)
We can see that the highest order derivative present in the differential is \( y^{\prime} \).
Thus, its order is one. It is polynomial equation in \(y^{\prime}\)
So, the highest power raised to \( y^{\prime} \) is \(1 \).
Therefore, its degree is one.
9. Determine order and degree (if defined) of differential equations given
\(y^{\prime \prime}+\left(y^{\prime}\right)+2 y=0\)
\(y^{\prime \prime}+\left(y^{\prime}\right)+2 y=0\)
Answer
It is given that equation is \( y^{\prime \prime}+\left(y^{\prime}\right)^{2}+2 y=0 \)
We can see that the highest order derivative present in the differential is \( y^{\prime \prime} \).
Thus, its order is two. It is polynomial equation in \( y^{\prime \prime} \) and \( y^{\prime} \)
So, the highest power raised to \( y^{\prime \prime} \) is \(1 \).
Therefore, its degree is one.
10. Determine order and degree (if defined) of differential equations given
\(y^{\prime \prime}+2 y^{\prime}+\sin y=0\)
\(y^{\prime \prime}+2 y^{\prime}+\sin y=0\)
Answer
It is given that equation is \( y^{\prime \prime}+2 y^{\prime}+\sin y=0 \)
We can see that the highest order derivative present in the differential is \(y^{\prime \prime} \).
Thus, its order is two. It is polynomial equation in \( y^{\prime \prime} \) and \( y^{\prime} \)
So, the highest power raised to \( y^{\prime \prime} \) is \(1 \).
Therefore, its degree is one.
11. The degree of the differential equation
\( \left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0 \) is
A. 3 B. 2 C. 1 D. not defined
\( \left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0 \) is
A. 3 B. 2 C. 1 D. not defined
Answer
It is given that equation is \( \left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0 \)
We can see that the highest order derivative present in the given differential equation is \( \frac{d^{2} y}{d x^{2}} \).
Thus, its order is three.
The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
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
A. 2 B. 1 C. 0 D. not defined
Answer
It is given that equation is \( 2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0 \)
We can see that the highest order derivative present in the given differential equation is \( \frac{d^{2} y}{d x^{2}} \).
Thus, its order is two.
