Solve these differential equations by converting to Clairaut's form

In summary, the given equations can be reduced to Clairaut's form by suitable substitutions, such as U = x and V = y or U = x + y and V = x + yp.
  • #1
vish_maths
61
1
The question comprises of three subparts which need to be converted to Clairaut's form and then solved :

(a) x p2 - 2yp + x + 2y = 0

(b) x2 p2 + yp (2x + y) + y2 = 0

(c) (x2+y2)(1+p)2-2(x+y)(1+p)(x+yp)+(x+yp)2=0

Note : p = dy/dx

I understand that Reducing to Clairaut's form involves suitable substitution so as to bring it in the form of V = P U + f(P) but i am unable to form any intuition about what such substitutions might be , as the above equations seem complicated with more than one combination of variables and 'p'.

I have added three sub-parts to get a better understanding of the intuition involved in such substitutions. Help would be greatly appreciated. Thanks.
 
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  • #2
Solution:(a) x p2 - 2yp + x + 2y = 0Let U = x and V = y. Then, the equation can be written as V = (U p2)/2 – U + 2This is in the form of Clairaut's equation.(b) x2 p2 + yp (2x + y) + y2 = 0Let U = x and V = y. Then, the equation can be written as V2 = U2 p2 + U p (2U + V)This is in the form of Clairaut's equation.(c) (x2+y2)(1+p)2-2(x+y)(1+p)(x+yp)+(x+yp)2=0Let U = x + y and V = x + yp. Then, the equation can be written as V2 = (U2 + V2)(1 + p)2 – 2UV(1 + p) + U2This is in the form of Clairaut's equation.
 

1. What is Clairaut's form?

Clairaut's form is a way of expressing a differential equation in the form of y = x * f(x) + g(x).

2. How do you convert a differential equation to Clairaut's form?

To convert a differential equation to Clairaut's form, you need to first differentiate the equation with respect to x, then substitute y' with -x * f'(x) + g'(x). This will give you an equation in the form of y = x * f(x) + g(x).

3. What are the advantages of using Clairaut's form to solve a differential equation?

Clairaut's form can make solving differential equations easier and more efficient, as it eliminates the need for integration and can lead to a simple algebraic equation.

4. Can all differential equations be converted to Clairaut's form?

No, not all differential equations can be converted to Clairaut's form. This form is only applicable to certain types of differential equations, such as those that are linear or can be transformed into a linear form.

5. Are there any other methods for solving differential equations?

Yes, there are multiple methods for solving differential equations, such as separation of variables, substitution, and using integrating factors. The best method to use will depend on the type of differential equation and the initial conditions given.

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