Differential Equations – Important Questions

A differential equation is an equation that includes the derivative of an unknown function. To determine how quickly a function changes at a given point, one must look at its derivatives. Through a differential equation, the derivatives of these functions are linked together. A differential equation can also be defined as a mathematical expression that describes the connection here between function and also its derivatives. When we examine y as a function of x, a differential equation is one that incorporates the derivatives of y with respect to x (or the differentials of y and x) with or without variables x and y.

Differential Equations – Important Questions

• Show that the differential equation (5x−8y).dy/dx = (4x+2y) is a homogeneous differential equation.

Solution:

(5x−8y).dy/dx = (4x+2y) is the given differential equation.

To prove that the above differential equation is homogeneous, let us substitute x =δ x and y =δ y.

Here we have F(x, y) = (4x+2y) / (5x−8y)

F(δx, δy) = ( δ4x+δ2y)/( δ5x− δ8y)

F(δx, δy) = δ(4x+2y) / δ(5x−8y) =  δ0 f(x, y)

Hence proved that the given equation is the homogeneous differential equation.

• Find the solution of the homogeneous differential equation xSin(y/x).dy/dx = ySin(y/x) + x.

Solution: The given differential equation is xSin(y/x).dy/dx = ySin(y/x) + x

dy/dx = {ySin(y/x) + x } / xsin(y/x)

dy/dx = {x((y/x).Sin(y/x)+1)} / xSin(y/x)

dy/dx = ((y/x).Sin(y/x)+1 /  sin(y/x)

Here, let us replace y/x =v in the above equation.

dy/dx = vSinv+1 / sinv 

Here, write y/x = v in the form of y = vx. On differentiating y = vx on both sides of the equation we obtain 

dy/dx = v + x.dv/dx, which is substituted in the above equation

v + x.dv/dx = vsinv + 1/ sinv     

x.dv/dx = (vSinv+ 1 / sin v) – v

Here we spilt the variables on both sides

x.dv/dx = 1/Sinv

Sinv.dv = dx/x

On integrating both equations, we get the below equation

∫sin v.dv=∫dx/x

-cos v = Logx + C

Here we switch back y/x = v.

-cos y/x = Logx + C

Therefore, the solution of the homogeneous differential equation is – cos y/x = Logx + C.

• Write examples of non-homogeneous differential equations.

Solution:

• d2ydx2 − 9 y = −6 cos 3 x 
• d3ydx3 + 2 dydx + x = 4 e –x
• d5ydx5   + 2yxdx3  + x = 4.
• d2ydx2 – 2 dydx+ 5 y = 10 xy − 3 x − 3
• d3ydx3  − 2 d2ydx2 + 5 y = 10 x2– 3y.
• d4ydx4  − 3 dydx = −12 x
•  d2ydx2  − 3dydx = −12 e –x

• Give stepwise approach to solve homogeneous differential equations.

Solution: We provide some of the easy steps to solve homogeneous differential equations.

Here is the given dydx=Fx,y=p(xy )

STEP 1 – For y, use y = vx in the given equation

STEP 2 – differentiate the y = vx we get dydx=v+x dvdx substitute the value in the equation

We get v+xdvdx=pv

xdvdx=pv-v.

STEP 3 –  if we separate the variable from the above equation we get

dvpv-v = dxx

STEP 4 – Integrate on both side of the equation we get

dvpv-v dv = dxx + c

STEP 5 – after completion of integration, replace v= y/x

Conclusion :

The connection between both the variables x and y that is produced after eliminating the derivatives (i.e., integration) or where the connection includes an arbitrary constant to signify the order of an equation is the basic solution of a differential equation. An arbitrary constant appears in the solution of first-order differential equations, while two arbitrary constants appear in the solutions of second-order difference equations. The solution of the differential equations is derived by assigning specific values to the arbitrary constant.