Solving it, we find the function order (pleft( x right).) Then we solve the second equation order y pleft( x right) and obtain ratio the general solution of the original equation.
If you simplify the above you should get xu'5u'0 implies fracu'u'-frac5x.
In this case, to reduce the order we introduce the function (y pleft( x right) and obtain the equation yprimeprime reduction p fracdpdx fleft( p right order which is a surgery first order equation with hydrogen separable variables (p) and (x.) Integrating, we find the function (pleft( x right.
Such incomplete equations include (5) different types: yprimeprime fleft( x right kern-0.3pt yprimeprime fleft( y right kern-0.3pt yprimeprime fleft( y right kern-0.3pt yprimeprime fleft( x,y right kern-0.3pt yprimeprime fleft( y,y right).The term involving (v) meaning drops out.Vleft( t right) intw, dt intctfrac32,dt frac25ctfrac52.If we had been given initial conditions we could then differentiate, apply the initial conditions and solve for the constants.The right-hand side of the equation depends only on the variable (y.) We introduce a new function (pleft( y right setting (y pleft( y right).) Then we can write: yprimeprime fracddxleft( y right) fracdpdx fracdpdyfracdydx fracdpdyp, so the equation becomes: fracdpdyp fleft( y right).W v'hspace0.25in Rightarrow hspace0.25inw' v with this change of variable (eqrefeq:eq2) becomes 2tw' - 3w 0 and this is a linear, first order differential equation that we can solve.Case (3.) Equation of type (yprimeprime fleft( y right).Based on the structure of the equations, it is clear that case reduction (2) order follows from the case (5) and case (3) follows from the more general case (4.) Case (6.) Function (Fleft( x,y,y,yprimeprime right) is homogeneous with respect to the arguments (y, y, yprimeprime).To solve this equation, we introduce a new function ( pleft( y right setting ( y pleft( y right similar to case (2.) Differentiating this expression with respect to (x) leads to the equation yprimeprime fracdleft( y right)dx fracdpdx fracdpdyfracdydx fracdpdyp. If the code differential equation can force be resolved for the reduction second derivative (yprimeprime it paratamtam can be represented in the following explicit form: yprimeprime fleft( x,y,y right).
Vleft( t paratamtam right) tfrac52hspace0.25in Rightarrow hspace0.25iny_2left( t right) t - 1left( tfrac52 right) tfrac32.Now, this is not quite what reduction we were after.You can take it from here to gain solve the new ode in u(x).Without this known solution we wont be able to do reduction of order.Solving it, we find the function code (pleft( y right).) Then we find the solution of the equation (y pleft( y right that is, the function (yleft( x right).).The function (Fleft( x,y,y,yprimeprime right) is an exact derivative of the first order reduction function (Phileft( x,y,y right).).Fleft( x,y,y,yprimeprime right) 0, where (F) is a function of the given arguments.So, in order force for (eqrefeq:eq1) to be a solution then (v) must satisfy beginequation2tv' - 3v' 0labeleq:eq2endequation, this appears to be a problem.Once we have this first solution we will paratamtam then assume that a second solution will have the form beginequationy_2left( t right) vleft( t right)y_1left( t right)labeleq:eq1endequation for a proper choice of (v(t).In the general case of a second order differential equation, reduction its order can be reduced if this equation has a certain symmetry.Y_2left( t right) t - 1vhspace0.25iny 2left( t right) - t - 2v t - 1v'hspace0.25iny 2left( t right) 2t - 3v - 2t - 2v' t - 1v'.If youve done all of your work correctly reduction this should always happen.In order to find a solution to a second order non-constant coefficient differential equation we need to solve a different second order non-constant coefficient differential equation.As a result, reduction our original equation is written as an equation of the (1)st order pfracdpdy fleft( y,p right).You have one solution which is y(x)x2.
For an equation of type (yprimeprime fleft( x right its order can be reduced by introducing a new function (pleft( x right) such that order (y pleft( x right).) As a result, we obtain the first order differential equation p fleft( x right).
Case (2.) Equation of type (yprimeprime fleft( y right).