Let $y(x) = a(x + 1) \log(x + 1) + bx + 5$ be the solution of the differential equation e$^\frac{dy}{dx} = x + 1_{;}y(0) = 5$, then the value of $(a + b)$ is:

Which of the following differential equations has $y = x$ as one of its particular solution?

The differential equation representing family of circles with centre at origin and radius r units is-

The general solution of the differential equation $\tan\left(\dfrac{(1+x^2)y'+2xy}{2x}\right)=1+x^2$, is

(Consider $\log_e x=\log x$)

(where C is an arbitrary constant)

Let $y(x)$ be the solution curve of the differential equation

$\frac{dy}{dx} + 2y^2 = 0, y(1) = 1$, then $y\left(\frac{3}{4}\right)$ is equal to:

Integrating factor of the differential equation $x\log x\frac{dy}{dx}+y=2\log x$ is equal to:

(Consider $\log x=\log_e x$)

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)