Differential Equations Quiz

The value of a for which the differential equation is exact is : (xy^2 +a x^3y^2)dx +(x^3y+yx)x dy=0

• 1

• -1

• 2

• None of these

Consider the differential equation: (x^2−y^2)dx−2xy dy= 0 Find the general solution.

• x^3−3xy^2=C

• x^2−y^2=C

• 3x^2y−y^3=C

• x^3+y^3=C

If the exact differential equation: M dx+N dy= 0 is solved using the method of integrating factors, then the integrating factorμ(x,y) is chosen such that:

• μ(x,y)M dx+μ(x,y)N dy satisfies the exactness condition.

• μ(x,y) always depends only on x.

• μ(x,y) always depends only on y.

• μ(x,y) must be constant.

The Particular Integral (PI) of the differential equation: (D^2−3D+ 2)y=e^x is:

• e^(x^2)

• e^x(1−3 + 2)

• e^x 0

• xe^x

The general form of a Cauchy-Euler equation of order n is:

• a_n D^n y+a_{n−1} D^{n−1} y+···+a_0 y= 0

• a_n x^n D^n y+a_{n−1} x^{n−1} D^{n−1} y+···+a_0 y= 0

• x^n y^{(n)}+a_{n−1} x^{n−1} y^{(n−1)}+···+a_0 y= 0

• x^n D^n y+x^{n−1} D^{n−1} y+···+y= 0

The substitution used to solve a Cauchy-Euler equation: x^2 y′′ + 3xy′ −4y= 0 is:

• y=e^x

• y= cosx

• y=x^m

• y=Ax+B

The formula for u′1 in Variation of Parameters is:

• u′1= - y2 f(x)/W(y1,y2)

• u′1=- y1 f(x)/W(y1,y2)

• u′1=y′2 f(x)/W(y1,y2)

• u′1=y′′2 f(x)/W(y1,y2)

Which of the following is a limitation of the Variation of Parameters method?

• It cannot be applied to equations with constant coefficients

• It cannot be applied when the Wronskian is zero

• It cannot be applied to second-order differential equations

• It requires the forcing function f(x) to be a polynomial

The complementary function (CF) of the differential equation (D^4−3D^2+ 2)y= 0 is:

• C1 e^x + C2 e^−x + C3 e^(Sqrt(2)x) + C4 e^(−Sqrt(2)x)

• C1 e^x + C2 e^−x + C3 e^(2x) + C4 e^(−2x)

• C1 e^(2x) + C2 e^(−2x) + C3 xe^x + C4 xe^−x

• C1 e^x + C2 xe^x + C3 e^−x + C4 xe^−x

What is the general solution of the differential equation: (D^2 + 4)y = 0?

• y = C1 + C2 x

• y = C1 e^x + C2 e^(-x)

• y = C1 e^(2x) + C2 e^(-2x)

• y = C1 cos(2x) + C2 sin(2x)