Exercise 9.1

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10.

${"font":{"family":"Arial","color":"#000000","size":10},"id":"1-0-0-0","code":"\\begin{align*}\n{1.\\,\\frac{d^{4}y}{dx^{4}}+\\sin\\left(y^{\\prime \\prime \\prime }\\right)}&={0}\t\n\\end{align*}","type":"align*","ts":1603613484488,"cs":"R9jQzLDGu9Jy/yzs8KwZEQ==","size":{"width":148,"height":36}}$

Solun:- Given eq. is:-

${"id":"1-1-0-0-0","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\frac{d^{4}y}{dx^{4}}+\\sin\\left(y^{\\prime \\prime \\prime }\\right)}&={0}\t\n\\end{align*}","type":"align*","ts":1603613471506,"cs":"a3Wv7KaeAig7HtNCfKEW/w==","size":{"width":132,"height":36}}$

${"font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\frac{d^{4}y}{dx^{4}}+\\sin\\left(\\frac{d^{3}y}{dx^{3}}\\right)}&={0}\t\n\\end{align*}","id":"2-0-0","ts":1603613522366,"cs":"wUntw0YdVdV1zXiuxsvGqA==","size":{"width":154,"height":38}}$

Order of the given differential equation is 4.

But the degree of the given differential equation is not defined because the given equation is not a polynomial equation of derivatives.

2. y’ + 5y = 0

Solun:- Given eq. is:- y’ + 5y = 0

${"code":"\\begin{align*}\n{\\frac{dy}{dx}+5y}&={0}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"1-1-1","ts":1603613859973,"cs":"spA0v0eIOaARSFVPAYFATg==","size":{"width":88,"height":32}}$

Order of the given differential equation is 1.

Degree of the given differential equation is also 1 because the degree of the differential equation is the power of the highest order derivative.

${"type":"align*","id":"1-0-1-0","code":"\\begin{align*}\n{3.\\,\\left(\\frac{ds}{dt}\\right)^{4}+3s.\\frac{d^{2}s}{dt^{2}}}&={0}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"ts":1603614371189,"cs":"K8ywGHatfnXJLweHKz3uDQ==","size":{"width":166,"height":40}}$

Solun:- Given eq. is:-

${"type":"align*","id":"3","code":"\\begin{align*}\n{\\left(\\frac{ds}{dt}\\right)^{4}+3s.\\frac{d^{2}s}{dt^{2}}}&={0}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1603614399918,"cs":"mSErSHfLGrEVMA/k8qY0tw==","size":{"width":152,"height":40}}$

Order of the given differential equation is 2.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

${"font":{"family":"Arial","size":10,"color":"#000000"},"id":"1-0-0-1-0","code":"\\begin{align*}\n{4.\\,\\left(\\frac{d^{2}y}{dx^{2}}\\right)^{2}+\\cos\\left(\\diff{y}{x}\\right)}&={0}\t\n\\end{align*}","type":"align*","ts":1603614659209,"cs":"Lz0FbMm14uPOTS0LjGY45Q==","size":{"width":196,"height":41}}$

Solun:- Given eq. is:-

${"type":"align*","id":"1-0-0-2","code":"\\begin{align*}\n{\\left(\\frac{d^{2}y}{dx^{2}}\\right)^{2}+\\cos\\left(\\diff{y}{x}\\right)}&={0}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1603614692376,"cs":"8y2/iEBDwl6LHQZhRrqU6Q==","size":{"width":180,"height":41}}$

Order of the given differential equation is 2.

But the degree of the given differential equation is not defined because the given equation is not a polynomial equation of derivatives.

${"type":"align*","code":"\\begin{align*}\n{5.\\,\\frac{d^{2}y}{dx^{2}}}&={\\cos3x+\\sin3x}\t\n\\end{align*}","id":"1-0-1-1-0","font":{"color":"#000000","family":"Arial","size":10},"ts":1603614876533,"cs":"zUHLv8PvUfdoxz8mO25AGw==","size":{"width":165,"height":34}}$

Solun:- Given eq. is:-

${"code":"\\begin{align*}\n{\\frac{d^{2}y}{dx^{2}}}&={\\cos3x+\\sin3x}\t\n\\end{align*}","id":"1-0-1-2","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","ts":1603614915622,"cs":"uHFMbEoz7Hk0wenHgURdTA==","size":{"width":150,"height":34}}$

Order of the given differential equation is 2.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

${"code":"\\begin{align*}\n{6.\\,\\left(y^{\\prime \\prime \\prime }\\right)^{2}+\\left(y^{\\prime \\prime }\\right)^{3}+\\left(y^{\\prime }\\right)^{4}+y^{5}}&={0}\t\n\\end{align*}","id":"1-0-1-1-1-0","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1603615424164,"cs":"ASNEV29CZB2UM/DQiD99FA==","size":{"width":222,"height":21}}$

Solun:- Given eq. is:-

${"id":"4-0-0","code":"\\begin{align*}\n{\\left(y^{\\prime \\prime \\prime }\\right)^{2}+\\left(y^{\\prime \\prime }\\right)^{3}+\\left(y^{\\prime }\\right)^{4}+y^{5}}&={0}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1603615446431,"cs":"fesXwFaxO0SmFGVGx8i/0g==","size":{"width":208,"height":21}}$

${"type":"align*","id":"4-1-0","code":"\\begin{align*}\n{\\left(\\frac{d^{3}y}{dx^{3}}\\right)^{2}+\\left(\\frac{d^{2}y}{dx^{2}}\\right)^{3}+\\left(\\diff{y}{x}\\right)^{4}+y^{5}}&={0}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1603615564292,"cs":"panKMApuD+tMIwq0o9eHDA==","size":{"width":276,"height":41}}$

Order of the given differential equation is 3.

Degree of the given differential equation is 2 because the degree of the differential equation is the power of the highest order derivative.

${"id":"1-0-1-1-1-1-0","code":"\\begin{align*}\n{7.\\,y^{\\prime \\prime \\prime }+2y^{\\prime \\prime }+y^{\\prime }}&={0}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"type":"align*","ts":1603615750669,"cs":"sKo1Qnl+dsHkCCM92wywHg==","size":{"width":134,"height":16}}$

Solun:- Given eq. is:-

${"code":"\\begin{align*}\n{y^{\\prime \\prime \\prime }+2y^{\\prime \\prime }+y^{\\prime }}&={0}\t\n\\end{align*}","type":"align*","id":"1-0-1-1-1-2-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1603615769846,"cs":"KebyvProvnwukSt6fmy+aw==","size":{"width":120,"height":16}}$

${"code":"\\begin{align*}\n{\\frac{d^{3}y}{dx^{3}}+2.\\frac{d^{2}y}{dx^{2}}+\\diff{y}{x}}&={0}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","id":"4-1-1-0","ts":1603615818115,"cs":"njq2wFSB/+EdhhrUB9/9Ew==","size":{"width":165,"height":34}}$

Order of the given differential equation is 3.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

${"code":"\\begin{align*}\n{8.\\,y^{\\prime }+y}&={e^{x}}\t\n\\end{align*}","id":"1-0-1-1-1-1-1-0","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1603615957779,"cs":"GXutNDO5/bLGtTJJbpNZZg==","size":{"width":88,"height":16}}$

Solun:- Given eq. is:-

${"id":"1-0-1-1-1-1-2-0","code":"\\begin{align*}\n{y^{\\prime }+y}&={e^{x}}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","ts":1603615997716,"cs":"4Ld07wmyVlYw2CDAlUR1EA==","size":{"width":73,"height":16}}$

${"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\diff{y}{x}+y}&={e^{x}}\t\n\\end{align*}","id":"4-1-1-1-0","type":"align*","ts":1603616070889,"cs":"sAWoMhQLwWba2jfKNrHe9Q==","size":{"width":88,"height":32}}$

Order of the given differential equation is 1.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

${"font":{"color":"#000000","family":"Arial","size":10},"id":"1-0-1-1-1-1-1-1-0","code":"\\begin{align*}\n{9.\\,y^{\\prime\\prime }+\\left(y^{\\prime }\\right)^{2}+2y}&={0}\t\n\\end{align*}","type":"align*","ts":1603616254387,"cs":"VOyjHyf2GZ/qFGIKsxgBvA==","size":{"width":145,"height":21}}$

Solun:- Given eq. is:-

${"type":"align*","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{y^{\\prime\\prime }+\\left(y^{\\prime }\\right)^{2}+2y}&={0}\t\n\\end{align*}","id":"1-0-1-1-1-1-1-2","ts":1603616277457,"cs":"5d9pYQelXoeC49dAkkcTAg==","size":{"width":130,"height":21}}$

${"type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\frac{d^{2}y}{dx^{2}}+\\left(\\diff{y}{x}\\right)^{2}+2y}&={0}\t\n\\end{align*}","id":"4-1-1-1-1-0","ts":1603616363127,"cs":"+22NNk/PhspC/x6ZITbQUw==","size":{"width":168,"height":40}}$

Order of the given differential equation is 2.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

${"code":"\\begin{align*}\n{10.\\,y^{\\prime\\prime }+2y^{\\prime }+\\sin y}&={0}\t\n\\end{align*}","id":"1-0-1-1-1-1-1-1-1","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"ts":1603616448285,"cs":"1HWVYV+6hlXTxl3N+gN3WQ==","size":{"width":154,"height":16}}$

Solun:- Given eq. is:-

${"id":"1-0-1-1-1-1-1-1-2","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{y^{\\prime\\prime }+2y^{\\prime }+\\sin y}&={0}\t\n\\end{align*}","ts":1603616498472,"cs":"A8nkHGVQF4Jj0HKejYggCw==","size":{"width":132,"height":16}}$

${"id":"4-1-1-1-1-1","type":"align*","code":"\\begin{align*}\n{\\frac{d^{2}y}{dx^{2}}+2.\\diff{y}{x}+\\sin y}&={0}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1603616528690,"cs":"roAU9bshzAJDZyjjtPDIgw==","size":{"width":164,"height":34}}$

Order of the given differential equation is 2.

Degree of the given differential equation is 1 because the degree of the differential equation is the power of the highest order derivative.

11. The degree of the differential equation

${"code":"\\begin{align*}\n{\\left(\\frac{d^{2}y}{dx^{2}}\\right)^{3}+\\left(\\diff{y}{x}\\right)^{2}+\\sin\\left(\\diff{y}{x}\\right)+1}&={0}\t\n\\end{align*}","id":"1-0-0-1-1","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","ts":1603616664149,"cs":"B87ejcCvuD4qiRbhaVHbTw==","size":{"width":280,"height":41}}$

(A) 3 (B) 2 (C) 1 (D) not defined

Solun:- Given eq. is:-

${"code":"\\begin{align*}\n{\\left(\\frac{d^{2}y}{dx^{2}}\\right)^{3}+\\left(\\diff{y}{x}\\right)^{2}+\\sin\\left(\\diff{y}{x}\\right)+1}&={0}\t\n\\end{align*}","id":"5","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","ts":1603616664149,"cs":"4wNUoFl20693JiHZKNAy8A==","size":{"width":280,"height":41}}$

Degree of the given differential equation is not defined because the given equation is not a polynomial equation of derivatives.

The correct answer is D.

12. The order of the differential equation

${"id":"1-0-0-1-1","code":"\\begin{align*}\n{2x^{2}\\frac{d^{2}y}{dx^{2}}-3\\diff{y}{x}+y}&={0}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","ts":1603616873495,"cs":"ot5a2L5FafSZ3dWNBt2P6w==","size":{"width":161,"height":34}}$

(A) 2 (B) 1 (C) 0 (D) not defined

Solun:- Given eq. is:-

Order of the given differential equation is 2.

The correct answer is A.

Download PDF of Exercise 9.1

NCERT Maths solutions for 12th

Important Note

Exercise 9.1