State True and False for the following (i) Integrating factor of the differential of the form (dx)/(dy)+p_(1)x=Q_(1) is given bye^(intP_(1)dy). (ii) Solution of the differential equation of the type (dx)/(dy)+P_(1)x=Q_(1) is given by x*IF=int(IF)xxQ_(1)dy. (iii) Correct substitution for the solution of the differential equation of the type (dy)/(dx)=f(x,y), where f(x, y) is homogeneous function of zero degree is y = vx. (iv) Correct substitution for the solution of the differential equation of the type (dy)/(dx)=g(x,y), where g(x,y) is a homogeneous function of the degree zero is x=vy. (v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (vi) The differential equation representing the family of circles x^(2)+(y-a)^(2)=a^(2) will be of order two. (vii) The solution of (dy)/(dx)=((y)/(x))^(1//3)"is "y^(2//3)-x^(2//3)=c (viii) Differential equation representing the family of curve y=e^(x)(Acosx+Bsinx)"is "(d^(2)y)/(dx^(2))-2(dy)/(dx)+2y=0. (ix) The solution of the differential equation (dy)/(dx)=(x+2y)/(x)"is "x+y=kx^(2). (x) Solution of (xdy)/(dx)=y+xtan""(y)/(x)"is "sin((y)/(x))=cx (xi) The differential equation of all non horizontal lines in aplane is (d^(2)x)/(dy^(2))=0.

State True and False for the following (i) Integrating factor of the

1.	State True and False for the following (i) Integrating factor of the differential of the form (dx)/(dy)+p_(1)x=Q_(1) is given bye^(intP_(1)dy). (ii) Solution of the differential equation of the type (dx)/(dy)+P_(1)x=Q_(1) is given by x*IF=int(IF)xxQ_(1)dy. (iii) Correct substitution for the solution of the differential equation of the type (dy)/(dx)=f(x,y), where f(x, y) is homogeneous function of zero degree is y = vx. (iv) Correct substitution for the solution of the differential equation of the type (dy)/(dx)=g(x,y), where g(x,y) is a homogeneous function of the degree zero is x=vy. (v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (vi) The differential equation representing the family of circles x^(2)+(y-a)^(2)=a^(2) will be of order two. (vii) The solution of (dy)/(dx)=((y)/(x))^(1//3)"is "y^(2//3)-x^(2//3)=c (viii) Differential equation representing the family of curve y=e^(x)(Acosx+Bsinx)"is "(d^(2)y)/(dx^(2))-2(dy)/(dx)+2y=0. (ix) The solution of the differential equation (dy)/(dx)=(x+2y)/(x)"is "x+y=kx^(2). (x) Solution of (xdy)/(dx)=y+xtan""(y)/(x)"is "sin((y)/(x))=cx (xi) The differential equation of all non horizontal lines in aplane is (d^(2)x)/(dy^(2))=0.
Answer» Solution :(i) True Given differential equation, `""(dx)/(dy)+P_(1)x=Q_(1)` `therefore""IF=e^(intp_(1)dy)` (ii) True (iii) True (IV) True (v) False There is no arbitrary constant in the particular solution of a differential equation. (vi) False We know that, order of the differential equation =number of arbitrary constant Here, number of arbitrary constant = 1 So order is one. (vii) True Given differential equation, `""(dy)/(dx) =((y)/(x))^(1//3)` `RARR""(dy)/(dx) =(y^(1//3))/(x^(1//3))` `rArr""y^(-1//3)dy=x^(-1//3)dx` On integrating both sides, we get `""inty^(-1//3)dy=intx^(-1//3)dx` `rArr""(y^(-1//3+1))/((-1)/(3)+1)=(x^(-1//3+1))/((-1)/(3)+1)+C'` `rArr""(3)/(2)y^(2//3)=(3)/(2)x^(2//3)+C'` `rArr""y^(2//3)-x^(2//3)=C'""["where",(2)/(3)C'=C]` (viii) True Given that, `""y=e^(x)(Acosx+Bsinx)` On differentiating w.r.t. x, we get `""(dy)/(dx)=e^(x)(-Asinx+Bcosx)+e^(x)(Acosx+(Bsinx)` `rArr""(dy)/(dx)-y=e^(x)(-Asinx+Bcosx)` Again differentiating w.r.t. x, we get `""(d^(2)y)/(dx^(2))-(dy)/(dx)=e^(x)(-Acosx-Bsinx)+e^(x)(-Asinx-Bcosx)+e^(x)(-Asinx+Bcosx)` `rArr""(d^(2)y)/(dx^(2))-(dy)/(dx)+y=(dy)/(dx)-y` `rArr""(d^(2)y)/(dx^(2))-2(dy)/(dx)+2y=0` (ix) True Given that, `""(dy)/(dx)=(x+2y)/(x)rArr(dy)/(dx)=1+(2)/(x)y` `rArr""(dy)/(dx)-(2)/(x)y=1` `""IF=e^((-2)/(x)dx)=e^(-2logx)=x^(-2)` The differential solution, `""yx^(-2)=intx^(-2)*1dx+k` `rArr""(y)/(x^(2))=(x^(-2+1))/(-2+1)+k` `rArr""(y)/(x^(2))=(-1)/(x)+k` `rArr""y=-x+kx^(2)` `rArr""x+y=kx^(2)` (x) True Given differential equation, `""(xdy)/(dx)=y+xtan((y)/(x))` `rArr""(dy)/(dx)=(y)/(x)+tan((y)/(x))""`...(i) ltBrgt Put `""(y)/(x)=v" "i.e., y=vx` `rArr""(dy)/(dx)=v+(xdv)/(dx)` On SUBSTITUTING these values in Eq. (i), we get `""(xdv)/(dx)+v=v+tanv` `rArr""(dx)/(x)=(dv)/(tanv)` On integrating both sides, we get `""int(1)/(x)dx=int(1)/(tanv)dx` `rArr""log(x)=log(sinv)+logC'` `rArr""log((x)/(sinv))=logC'` `rArr""(x)/(sinv)=C'` `rArr""sinv=Cx""["where, C=(1)/(C')]` `rArr""sin(y)/(x)=Cx` (XI) True Let any non-horizontal line in a plane is given by `""y=mx+c` On differentiating w.r.t. x, we get `""(dy)/(dx)=m` Again, differentiating w.r.t. x, we get `""(d^(2)y)/(dx^(2))=0`

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