Solve the Differential Equation

Question

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Solution

`y+x dy/dx=x−y dy/dx`

`x dy/dx + y dy/dx=x−y`

`⇒dy/dx=(x−y)/(x+y) ` ......(1)

`Let F(x, y) =(x−y)/(x+y)`

`F(λx, λy) = λF(x, y)`
Therefore, F(x, y) is a homogeneous function of degree zero.

Let `y=vx`

`dy/dx=v+x (dv)/dx`

Substituting the value of y and dy/dx in (1) we get,

`v + x (dv)/dx=(x−vx)/(x+vx)=(1−v)/(1+v)`

`x (dv)/dx=(1−v)/(1+v)−v=(1−v−v^2−v)/(1+v)=(1−2v−v^2)/(1+v)`

`(1+v)/(v^2+2v−1)dv=−dx/x`

Integrating both sides, we have

`1/2 log∣(y^2/x^2)+(2y)/x−1∣+log|x|=logc`

`⇒log∣(y^2/x^2)+(2y)/x−1∣+2log|x|=2logc`

`⇒log((y^2/x^2)+(2y)/x−1)(x^2)=logc^2`

`⇒((y^2+2yx−x^2)/x^2)(x^2) = c^2`

`⇒y^2+2yx−x^2=C (where C=c^2)`

shaalaa.com

Methods of Solving Differential Equations> Homogeneous Differential Equations

Is there an error in this question or solution?

Q 17 Q 16 Q 18.1

2015-2016 (March) All India Set 2 C

APPEARS IN

2015-2016 (March) All India Set 2 C (with solutions)

Q 17 | 4 marks

Video TutorialsVIEW ALL [3]

RELATED QUESTIONS

Find the particular solution of the differential equation:

2y e^x/y dx + (y - 2x e^x/y) dy = 0 given that x = 0 when y = 1.

Show that the given differential equation is homogeneous and solve them.

(x² + xy) dy = (x² + y²) dx

A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.

Which of the following is a homogeneous differential equation?

Prove that x² – y² = c (x² + y²)² is the general solution of differential equation (x³ – 3x y²) dx = (y³ – 3x²y) dy, where c is a parameter.

\[xy \log\left( \frac{x}{y} \right) dx + \left\{ y^2 - x^2 \log\left( \frac{x}{y} \right) \right\} dy = 0\]

\[\left( x^2 + y^2 \right)\frac{dy}{dx} = 8 x^2 - 3xy + 2 y^2\]

\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]

\[x \cos\left( \frac{y}{x} \right) \cdot \left( y dx + x dy \right) = y \sin\left( \frac{y}{x} \right) \cdot \left( x dy - y dx \right)\]

Solve the following initial value problem:
(xy − y²) dx − x² dy = 0, y(1) = 1

Solve the following initial value problem:
(y⁴ − 2x³ y) dx + (x⁴ − 2xy³) dy = 0, y (1) = 1

Solve the following initial value problem:
x (x² + 3y²) dx + y (y² + 3x²) dy = 0, y (1) = 1

Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]

Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution

Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`