A thermometer reading 800P is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50^of At any time t the thermometer reading be TOP and the outside temperature be S^oF.
Based on the above information, answer the following questions.
(i) If \(\lambda\) is positive constant of proportionality, then \(\frac{d T}{d t}\) is

(ii) The value of T(S) is

(iii) The value of T(10) is

(iv) Find the general solution of differential equation formed in given situation.

(v) Find the valiie of constant of integration c in the solution of differential equation formed in given situation.

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective measured the body temperature and found it to be 70°F. Two hours later, the detective measured the body temperature again and found it to be 60°F, where the room temperature is 50°F. Also, it is given the body temperature at the time of death was normal, i.e., 98.6°F.
Let T be the temperature of the body at any time t and initial time is taken to be 8 p.m.
 
Based on the above information, answer the following questions.
(i) By Newton's law of cooling,\(\frac{d T}{d t}\) is proportional to

(ii) When t = 0, then body temperature is equal to

(iii) When t = 2, then body temperature is equal to

(iv) The value of T at any time tis

(v) If it is given that loge(2.43) = 0.88789 and loge(0.5) = -0.69315, then the time at which the murder occur is

Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation.
Degree : The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation
in derivatives for the degree to be defined.
Based on the above information, answer the following questions.
(i) Find the degree of the differential equation \(2 \frac{d^{2} y}{d x^{2}}+3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}=0\) 

(ii) Order and degree of the differential equation \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\) are respectively

(iii) Find order and degree of the equation \(y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0\) 

(iv) Determine degree of the differential equation \((\sqrt{a+x}) \cdot\left(\frac{d y}{d x}\right)+x=0\) 

(v) Order and degree of the differential equation \(\left(1+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{7}{3}}=7 \frac{d^{2} y}{d x^{2}}\) are respectively

A differential equation is said to be in the variable separable form if it is expressible in the form j(x) dx = g(y) dy. The solution of this equation is given by \(\int f(x) d x=\int g(y) d y+c\) where c is the constant of integration. 
Based on the above information, answer the following questions.
(i) If the solunon of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\) represents a circle, then the value of' a 'is

(ii) The diiftfterential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}\) deterrnines a family of circle with 

(iii) If y' = y + 1, y (0) = 1, theny (In 2) =

(iv) The solution of the differential equation \(\frac{d y}{d x}=e^{x-y}+x^{2} e^{-y}\) is

(v) If \(\frac{d y}{d x}=y \sin 2 x, y(0)=1\) then its solution is

If an equation is of the form \(\frac{d y}{d x}+P y=Q\) ,where P, Q are functions of x, then such equation is known as linear differential equation. Its solu~:n is given by \(y \cdot(\mathrm{I} . \mathrm{F} .)=\int \mathrm{Q} \cdot(\mathrm{I} . \mathrm{F} .) d x+c\) where \(\text { I.F. }=e^{\int P d x}\) .
Now, suppose the given equation is \((1+\sin x) \frac{d y}{d x}+y \cos x+x=0\) 
Based on the above information, answer the following questions
(i) The value of P and Q respectively are

(ii) The value of I.F. is

(iii) Solution of given equation is

(iv) If y(0) = 1, then y,equals

(v) Value of \(y\left(\frac{\pi}{2}\right)\) is