( xy ≠ ) . y – cos y = x : ( y sin y + cos y + x ) y ′ = y . x + y = tan – y : y y ′ + y + = . y = a x ∈ (– a , a ) : x + y dy dx = ( y ≠ ) .
The number of arbitrary constants in the general solution of a differential equation of fourth order are: (A) (B) (C) (D) . The number of arbitrary constants in the particular solution of a differential equation of third order are: (A) (B) (C) (D) . . Methods of Solving First Order, First Degree Differential Equations In this section we shall discuss three methods of solving first order first degree differential equations.
. . Differential equations with variables separable A first order-first degree differential equation is of the form dx = F( x , y ) If F( x , y ) can be expressed as a product g ( x ) h ( y ), where, g ( x ) is a function of x and h ( y ) is a function of y , then the differential equation ( ) is said to be of variable separable type. The differential equation ( ) then has the form dx = h ( y ) .
g ( x ) If h ( y ) ≠ , separating the variables, ( ) can be rewritten as ( ) h y dy = g ( x ) dx ... ( ) Integrating both sides of ( ), we get ( ) dy h y ( ) g x dx ... ( ) Thus, ( ) provides the solutions of given differential equation in the form H( y ) = G( x ) + C Here, H ( y ) and G ( x ) are the anti derivatives of ( ) h y and g ( x ) respectively and C is the arbitrary constant. Example Find the general solution of