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i) Integrating factor of the differential of the form $\frac{d x}{d y}+P_{1} x=Q_{1}$ is given by $e^{\int P_{1}d y}$

(ii) Solution of the differential equation of the type $\frac{d x}{d y^{\prime}}+P_{1} x=Q_{1}$ is given by $x e^{\int P_{1} d y}=\int Q_{1} e^{\int P_{1} d y} d y+C$

iii) Correct substitution for the solution of the differential equation of the type $\frac{d y}{d x} f(x, y), where f(x, y)$ is a homogeneous function of zero degree is y=v x
(iv) Correct substitution for the solution of the differential equation of the type $\frac{d x}{d y} g(x, y)$ where g(x, y) is a homogeneous function of the degree zero is x=v y
(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 $\frac{d y}{d x}=\left(\frac{y}{x}\right)^{1 / 3} is y^{\frac{2}{3}}-x^{\frac{2}{3}}=c$

(viii) Differential equation representing the family of curves $y=e^{x}(A \cos x+B \sin x ) is \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$

ix) The solution of the differential equation $\frac{d y}{d x}=\frac{x+2 y}{x} is x+y=k x^{2}.$

x) Solution of   $\frac{x d y}{d x}=y+x \tan \left(\frac{y}{x}\right) \text { is } \sin \frac{y}{x}=C x$

xi) The differential equation of all non horizontal lines in a plane is   $\begin{aligned} &\text { }\\ &\frac{d^{2} x}{d y^{2}}=0 \end{aligned}$

Answers (1)

i) Integrating factor of the differential of the form $\frac{d x}{d y}+P_{1} x=Q_{1}$ is given by $e^{\int P_{1}d y}$ . Hence given statement is true.

(ii) Solution of the differential equation of the type $\frac{d x}{d y^{\prime}}+P_{1} x=Q_{1}$ is given by $x e^{\int P_{1} d y}=\int Q_{1} e^{\int P_{1} d y} d y+C$ .

Hence given statement is true.

iii) Correct substitution for the solution of the differential equation of the type $\frac{d y}{d x} f(x, y),$ where $f(x, y)$ is a homogeneous function of zero degree is y=v x.

Hence given statement is true.

(iv) Correct substitution for the solution of the differential equation of the type $\frac{d x}{d y} g(x, y)$ where g(x, y) is a homogeneous function of the degree zero is x=v y.

Hence given statement is true.

(V) There is no arbitrary constants in the particular solution of a differential equation. Hence given statement is Flase.

(vi) In thegiven equation $x^{2}+(y-a)^{2}=$ $a^{2}$ the number of arbitrary constant is one. So the order order will be one.