Question

The order of the differential equation whose general solution is
y = c1sin^3x + c2e^2x + c3 sin 3x + c4 sin x + c5e^2x+C6, is​

Answer 1

SOLUTION :

TO DETERMINE

The order of the differential equation whose general solution is

$\sf{}y = c_1 { \sin}^{3}x + c_2 {e}^{2x} + c_3 \sin 3x+ c_4 \sin x+ c_5 {e}^{2x} + c_6$

CONCEPT TO BE IMPLEMENTED

DIFFERENTIAL EQUATION

A differential equation is an equation which involves differential coefficients or differentials

ORDER OF A DIFFERENTIAL EQUATION

The order of a differential equation is the order of the highest derivative appearing in it.

DEGREE OF A DIFFERENTIAL EQUATION

The degree of a differential equation is the degree of the highest derivative occuring in it after the equation has been expressed in a form free from radicals and fractions as far as the derivatives are concerned

GENERAL SOLUTION OF DIFFERENTIAL EQUATION

A solution of a differential equation is a relation between the variables which satisfies the given differential equation.

RELATION BETWEEN ORDER AND NUMBER OF CONSTANTS

The general Solution of a differential equation is that in which the number of arbitrary constants is equal to the order of the differential equation.

EVALUATION

Here the general Solution of the differential equation is

$\sf{}y = c_1 { \sin}^{3}x + c_2 {e}^{2x} + c_3 \sin 3x+ c_4 \sin x+ c_5 {e}^{2x} + c_6 \: \: ..(1)$

Which can be rewritten as below

$\implies \sf{}y = c_1 { \sin}^{3}x + c_3 (3 \sin x - 4 { \sin}^{3} x)+ c_4 \sin x+ c_2 {e}^{2x} + c_5 {e}^{2x} + c_6$

$\implies \sf{}y =( c_1 - 4c_3) { \sin}^{3}x + (3c_3 + c_4 ) \sin x+ (c_2 + c_5) {e}^{2x} + c_6$

$\implies \sf{}y =A { \sin}^{3}x + B \sin x+ C {e}^{2x} + D \: \: ...(2)$

$\sf{Where \: A=c_1-4c_3, B=3c_3+c_4,C=c_2+c_5, D = c_6}$

Thus the given Solution in (1) contains six constants but the equation can rewritten as in (2) such that it contains four constants

Since The general Solution of the differential equation contains four arbitrary constants

So the order of the differential equation is 4

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The degree of the differential equation

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