Solving Homogeneous Differential Equation MCQ Quiz - Objective Question with Answer for Solving Homogeneous Differential Equation - Download Free PDF

Calculation

Given equation:

Divide by x:

Let y = vx, then

Substitute in the equation:

⇒ 

⇒ 

⇒ 

Integrate both sides:

⇒ 

⇒ 

Remove the logarithms:

⇒ 

Substitute v = y/x:

⇒ 

∴ The general solution is .

Hence option 2 is correct

Calculation

Let

⇒ 

⇒ 

Integrating both sides:-

⇒ 

⇒ 

Hence, option 1 is correct

Calculation

.....

Take,

The given equation becomes

Hence option 2 is correct

Calculation

Let

⇒ 

⇒ 

Integrating both sides:-

⇒ 

⇒ 

Hence, option 1 is correct

Calculation

⇒

Let , then

⇒

⇒

⇒

Integrating both sides:

⇒

⇒

⇒

⇒

⇒

Given , so

⇒

⇒

⇒

∴

Hence option 3 is correct

Concept:

Some useful formulas are:

Calculation:

⇒

Substituting y = vx and  

⇒ 

⇒ 

Integrating both sides we get,

⇒ , c = constant of integration

Putting the value of v we get,

∴ 

Concept:

Some useful formulas are:

Calculation:

Substituting y=vx and 

log x = log (sinv) + log c

x = c sinv

Putting the value of v we get,

[Where c = constant of integration]

Concept:

If a differential equation has the form f(x,y)dy = g(x,y)dx then it is said to be a homogeneous differential equation if the degree of f(x,y) and g(x, y) is the same.

Some useful formulas are;

Calculation:

Taking y = vx where  then we get

Integrating the above equation we get,

Now finally putting the value of (v = y/x) in the equation we get,

Concept:

If a differential equation has the form f(x,y)dy = g(x,y)dx then it is said to be a homogeneous differential equation if the degree of f(x,y) and g(x, y) is the same.

Some useful formulas are;

Calculation:

Taking y = vx where  then we get

Integrating the above equation we get,

Now finally putting the value of y in the equation we get,

Concept:

Some useful formulas are:

Calculation:

⇒

Substituting y = vx and  

⇒ 

⇒ 

Integrating both sides we get,

⇒ , c = constant of integration

Putting the value of v we get,

∴ 

Concept:

Some useful formulas are:

Calculation:

(xy + y2)dx = (x2 - xy)dy

Dividing both sides by x² we get,

Now substituting y = vx and 

Integrating the equation we get,

, c= constant of integration

Putting the value of v we get,

Concept:

Some useful formulas are:

Calculation:

Substituting y=vx and 

log x = log (sinv) + log c

x = c sinv

Putting the value of v we get,

[Where c = constant of integration]

Calculation:

x(x + y + z) = 9 ....(1)

y(x + y + z) = 16 ....(2)

z(x + y + z) = 144 .....(3)

By adding these three equations

x(x + y + z) + y(x + y + z) + z(x + y + z) = 9 + 16 + 144

⇒ (x + y + z)(x + y + z) = 169

⇒ (x + y + z)² = 169

⇒ (x + y + z) = 13

From eq. (1)

⇒ x(x + y + z) = 9

⇒ x(13) = 9

⇒ x = 

Concept:

Some useful formulas are:

Calculation:

⇒

Substituting y = vx and  

⇒ 

⇒ 

Integrating both sides we get,

⇒ , c = constant of integration

Putting the value of v we get,

∴ 

Given:

Concept:

To solve such type of differential equations, simply put y = vx.

Calculation:

Putting y = vx

⇒  

Now, 

⇒  = v + 

⇒  = 

Integrating both sides -

⇒ ln|f(v)| = ln|x| + lnC

⇒ f(v) = Cx

putting v = y/x back,

⇒ f(y/x) = Cx