Maharashtra Board Solutions for Chapter: Differential Equations, Exercise 5: EXERCISE 6.5

Solve the following differential equation:
$\frac{dy}{dx}+\frac{y}{x}=x^3-3$

Solve the following differential equation :

$\left(x+2y^3\right)\frac{dy}{dx}=y$

Solve the following differential equation:
$(x+y)\frac{dy}{dx}=1$

Solve the following differential equations:
$\left(x+a\right)\frac{dy}{dx}-3y={\left(x+a\right)}^5$

Solve the differential equation $ydx+\left(x-y^2\right)dy=0$

Find the equation of the curve passing through the point $\left(\frac{3}{\sqrt{2}},\sqrt{2}\right)$ having slope of the tangent to the curve at any point $(x, y)$ is $-\frac{4x}{9y}$.

The rate of growth of bacteria is proportional to the number present. If initially, there were $1000$ bacteria and the number doubles in $1$ hour, find the number of bacteria after $2\frac{1}{2}$ hours.

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are $25$ gms of certain substance and two hours later it is found that $9$ gms are left. Find the amount left after one more hour.