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NCERT · Class 12 · Maths

Chapter 9: Differential Equations

Practice each question, then open the answer or worked solution.

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Exercise 9.1

12 questions
Q1Written answer

Determine order and degree (if defined) of differential equation $\frac{{{\text{d}}^{\text{4}}}\text{y}}{\text{d}{{\text{x}}^{\text{4}}}}\text{+sin}\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)\text{=0}.$

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Worked solution

Ans: Rewrite the equation $\frac{{{\text{d}}^{\text{4}}}\text{y}}{\text{d}{{\text{x}}^{\text{4}}}}\text{+sin}\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)\text{=0}$. as:

$\Rightarrow \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ +sin}\left( y''' \right)=0$ 

The highest order between the two terms is of $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$ which is four.

The differential equation contains a trigonometric derivative term and is not completely polynomial in its derivative, thus degree is not defined.


Q2Written answer

Determine order and degree (if defined) of differential equation $\text{y }\!\!'\!\!\text{ +5y=0}$.

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Worked solution

Ans: The given differential equation is $\text{y }\!\!'\!\!\text{ +5y=0}$.

The highest order term is $\text{y }\!\!'\!\!\text{ }$, thus the order is one.

As the derivative is of completely polynomial nature is and highest power of derivative is of $\text{y }\!\!'\!\!\text{ }$ which is one. Thus degree is one.


Q3Written answer

Determine order and degree (if defined) of differential equation ${{\left( \frac{\text{ds}}{\text{dt}} \right)}^{\text{4}}}\text{+3s}\frac{{{\text{d}}^{\text{2}}}\text{s}}{\text{d}{{\text{t}}^{\text{2}}}}\text{=0}$.

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Worked solution

Ans: The given differential equation is ${{\left( \frac{\text{ds}}{\text{dt}} \right)}^{\text{4}}}\text{+3s}\frac{{{\text{d}}^{\text{2}}}s}{\text{d}{{\text{t}}^{\text{2}}}}\text{=0}$.

The highest order term is $\frac{{{\text{d}}^{\text{2}}}\text{s}}{\text{d}{{\text{t}}^{\text{2}}}}$, thus the order is two.

As the derivative is of completely polynomial nature is and highest power of derivative term$\frac{{{\text{d}}^{\text{2}}}\text{s}}{\text{d}{{\text{t}}^{\text{2}}}}$ which is one. Thus the degree is one.


Q4Written answer

Determine order and degree (if defined) of differential equation ${{\left( \frac{{{\text{d}}^{\text{2}}}\text{y}}{\text{d}{{\text{x}}^{\text{2}}}} \right)}^{\text{2}}}\text{+cos}\left( \frac{\text{dy}}{\text{dx}} \right)\text{=0}$.

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Worked solution

Ans: The given differential equation is ${{\left( \frac{{{\text{d}}^{\text{2}}}\text{y}}{\text{d}{{\text{x}}^{\text{2}}}} \right)}^{\text{2}}}\text{+cos}\left( \frac{\text{dy}}{\text{dx}} \right)\text{=0}$.

The highest order term is $\frac{{{\text{d}}^{\text{2}}}y}{\text{d}{{\text{x}}^{\text{2}}}}$, thus the order is two.

The differential equation contains a trigonometric derivative term and is not completely polynomial in its derivative, thus degree is not defined.


Q5Written answer

Determine order and degree (if defined) of differential equation ${{\left( \frac{{{\text{d}}^{\text{2}}}\text{y}}{\text{d}{{\text{x}}^{\text{2}}}} \right)}}\text{-cos3x+sin3x=0}$.

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Worked solution

Ans: The given differential equation is ${{\left( \frac{{{\text{d}}^{\text{2}}}\text{y}}{\text{d}{{\text{x}}^{\text{2}}}} \right)}^{\text{2}}}\text{-cos3x+sin3x=0}$.

The highest order term is $\frac{{{\text{d}}^{\text{2}}}y}{\text{d}{{\text{x}}^{\text{2}}}}$, thus the order is two.

As the derivative is of completely polynomial nature is and highest power of derivative term $\frac{{{\text{d}}^{\text{2}}}y}{\text{d}{{\text{x}}^{\text{2}}}}$ which is one. Thus degree is one.


Q6Written answer

Determine order and degree (if defined) of differential equation ${{\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)}^{\text{2}}}\text{+}{{\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)}^{\text{3}}}\text{+}{{\left( \text{y }\!\!'\!\!\text{ } \right)}^{\text{4}}}\text{+}{{\text{y}}^{\text{5}}}\text{=0}$.

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Worked solution

Ans: The given differential equation is ${{\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)}^{\text{2}}}\text{+}{{\left( \text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ } \right)}^{\text{3}}}\text{+}{{\left( \text{y }\!\!'\!\!\text{ } \right)}^{\text{4}}}\text{+}{{\text{y}}^{\text{5}}}\text{=0}$.

The highest order term is ${{\left( y''' \right)}^{2}}$, thus the order is three.

The differential equation is of the polynomial form and the power of highest order term $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$ is two, thus the degree is two.


Q7Written answer

Determine order and degree (if defined) of differential equation $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ +2y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +y }\!\!'\!\!\text{ =0}$.

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Worked solution

Ans: The given differential equation is $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{  }\!\!'\!\!\text{ +2y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +y }\!\!'\!\!\text{ =0}$.

The highest order derivative in the differential equation is $y'''$. Thus its order is three.

The differential equation is polynomial with the highest order term $y'''$ having a degree one. Thus the degree is one.


Q8Written answer

Determine order and degree (if defined) of differential equation $\text{y }\!\!'\!\!\text{ +y=e }\!\!'\!\!\text{ }$.

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Worked solution

Ans: The given differential equation is $\text{y }\!\!'\!\!\text{ +y=e }\!\!'\!\!\text{ }$. Therefore:

$\Rightarrow \text{y }\!\!'\!\!\text{ +y-e }\!\!'\!\!\text{ =0}$ 

The highest order derivative in the differential equation is $\text{y }\!\!'\!\!\text{ }$. Thus its order is one.

The given equation is of polynomial form with the highest order term $\text{y }\!\!'\!\!\text{ }$ with degree one. Thus the degree is one.


Q9Written answer

Determine order and degree (if defined) of differential equation $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +}{{\left( \text{y }\!\!'\!\!\text{ } \right)}^{\text{2}}}\text{+2y=0}$.

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Worked solution

Ans: The given differential equation is $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +}{{\left( \text{y }\!\!'\!\!\text{ } \right)}^{\text{2}}}\text{+2y=0}$.

The highest order derivative in the differential equation is $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$. Thus its order is two.

The given equation is of polynomial form with the highest order term $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$ with highest degree one. Thus the degree is one.


Q10Written answer

Determine order and degree (if defined) of differential equation $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +2y }\!\!'\!\!\text{ +siny=0}$ .

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Worked solution

Ans: The given differential equation is $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ +2y }\!\!'\!\!\text{ +siny=0}$.

The highest order derivative in the differential equation is $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$. Thus its order is two.

The given equation is of polynomial form with the highest order term $\text{y }\!\!'\!\!\text{  }\!\!'\!\!\text{ }$ with the highest degree one. Thus the degree is one.