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1.

Verify that `y=log(x+sqrt(x^2+a^2))` satisfies the differential equation `d^2y/(dx^2)+x(dy)/(dx)=0`

Answer» `y=log"("x+sqrt(x^(2)+a^(2))")"`
`implies(dy)/(dx)=(1)/((x+sqrt(x^(2)+a^(2)))).{1+(2x)/(2sqrt(x^(2)+a^(2)))}impliesy_(1)=(1)/(sqrt(x^(2)+a^(2)))`
`impliesy_(1)^(2)(x^(2)-a^(2))=1impliesy_(1)^(2)(2x)+(x^(2)+a^(2))2y_(1)y_(2)=0`
`(x^(2)+a^(2))y_(2)+xy_(1)=0.`
Discussion
2.

Verify that `y = A cos x - Bsin x` is a solution of the differential equation `(d^2y)/(dx^2)+y=0`

Answer» Given: `y=A cos x-B sin x`
`implies(dy)/(dx)=-A sinx-B cos x`
`implies (d^(2)y)/(dx^(2))=-Acosx+Bsinx`
`=-(A cosx-Bsinx)=-y" "["from (i)"]`
`implies(d^(2)y)/(dx^(2))+y=0.`
Hence, `y=Acos x-Bsinx` is a solution of the differential equation `(d^(2)y)/(dx^(2))+y=0.`
Discussion
3.

Writethe differential equation representing the family of curves `y=m x ,`where m is an arbitrary constant.

Answer» The equation of the given family of curves is `y=mx" "…(i),` where m is constant. Since the given equation contains one arbitrary constant, we differentiate it once only. On differentiating (i) w.r.t.x, we get
`(dy)/(dx)=m" "…(ii)`
Putting this value of m from (ii) in (i), we get
`y=((dy)/(dx))ximplies((dy)/(dx))-y=0.`
Hence, `x((dy)/(dx))-y=0` is the required differential equation.
Discussion
4.

Write the order and the degree of the differential equation `(d^(2)y)/(dx^(2))+5(dy)/(dx)+3y=0.`

Answer» In the giiven equation, the highest-order dervatives is `(d^(2)y)/(dx^(2))` and its power is 1.
`therefore` its order = 2 and degree=1.
Discussion
5.

Writhe the order and degree of the differentialequation `y=x(dy)/(dx)+a sqrt(1+((dy)/(dx))^2)dot`

Answer» The given equation may be weitten as
`sqrt(1+((dy)/(dx))^(2))+(y-x(dy)/(dx))`
`implies 1+((dy)/(dx))^(2)=(y-x(dy)/(dx))^(2)" "["on squaring both sides"]`
`implies1+((dy)/(dx))^(2)=y^(2)+x^(2)((dy)/(dx))-2xy(dy)/(dx)`
`implies(1-x^(2))((dy)/(dx))^(2)+2xy(dy)/(dx)+(1-y^(2))=0.`
Clearly, it is a differential equation of order = 1 and degree =2.
Discussion
6.

Write order and degree (if defined) of each of the following differential equations. `(dy)/(dx)+sin((dy)/(dx))=0`

Answer» Correct Answer - 1, not defined
Discussion
7.

Write order and degree (if defined) of each of the following differential equations. `((dy)/(dx))^(4)+3y((d^(2)y)/(dx^(2)))=0`

Answer» Correct Answer - `2,1`
Discussion