6. The differential equation (1+x²)-2xy = x + x³ has the general solution(A) y=e+²(x+x¹) + C4(B) y = e~²ln(1 +x²) + C(X2² +*²)(C) y=x²y+C₁x² + C₂x4(D) y=(1+x²) ln(1+x²) + C(1+x²)(E) y = Ce*² (²+¹)

The given problem is to find the general solution for the given differential equation.

Answered: dy 6. The differential equation (1+x²)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn.dydx+P(x)y=Q(x)yn. Observe that, if n=0n=0 or 11, the Bernoulli equation is linear. For other values of nn, the A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn.dydx+P(x)y=Q(x)yn. Observe that, if n=0n=0 or 11, the Bernoulli equation is linear. For other values of nn, the substitution u=y1−nu=y1−n transforms the Bernoulli equation into the linear equation dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x). Use an appropriate substitution to solve the equation y′−(5/x)y=y^4/x^10,y′−((5/x)y)=y^4/x^13, and find the solution that satisfies y(1)=1.y(1)=1.
5. The exact equation (3x2y+e')dx + (x³ + xey-2y)dy = 0 has the general solution x²+x2² - 2xy 4 (A) x³y+xey-y² = C x³ + xey *+2²2²-2xy (C) y = x4 4 (E) y = ln(x³ +)+C (B) y=x3 +xe? + (D) 如 x³ + xey x3 x4 +/+*2²2 - 2xy 4 =C
The Differential equation dx + 2x (y +1). dy (y + 1)³/2 is equivalent to Select one: d а. - (x(1+ y)²) = (y + 1)³/2 dy %3D b. d (y(1+ x)²) = (y +1)5/2 dy c. None of these d. d (æ(1+ y)²) = (y + 1)5/2 dx d (x(1+ y)²) = (y +1)5/2 dy е. %3D
For the differential equation (1 − 2x)y'' + 2y' + (2x − 3)y = e−x (a) Write the associated homogeneous equation and check that y1(x) = e x and y2(x) = xe−x are two linearly independent solutions of the associated equations. (b) Check that yp(x) = − 1/4 e−x is a solution of the full equation. (c) Use superposition principle to write down the general solution of the full equation. (d) Use the existence theorem or the Wronskian to find the intervals on which the solution is valid.
For the differential equation (2y - x³) dx + 3 x y'dy = 0, what is P (x) and q ( x) ? 2 А р (х) %3D ; q (x) = 3x 3 -2 -; q (x) = 3x B p ( x) = 3 x2 -3 (с) р(х) %3 -; q (x) = - 3 x² 2x 3 -; q (x) = 2.x p (x) = 3
(ii) Eliminate the arbitrary constant in the equation y = x² + ₁²x + c₂e³x Answer must be in the form d²y + a₁(x). dxn a₁(x). dn-ly dxn-1 dy + ... an-1(x). + an(x)y = R(x) dx
For the differential equation (2y3 – x³) dx + 3x y²dy = 0, what is p (x) and q (x) ? A p(x) B = p(x) = = p(x) = D) p (x) 3 - -; q (x) 2x 2 3x ; q (x) = = -x² 3 2 X - 3 -3 12/²³ ; 9(x) = -; q (x) = - 3x² 2x -2 = ²: 9 (x) = 3x² 3x
The General solution of the differential equation 3x2 + 2x + 6 is x3 + x2 + 6x dy using variable separable dx O y = x3 + x? + 6x + c O y = ln (x³ + x² + 6x) + c 3 O y = ln (x³ + x² + 6x) O y = In (x2 + x² + 6x) + c 3
The particular solution of the differential equation y +y- 2y = sin3x is: y(x)=Cje3*+Ce-3x y(x)=Cje2x+cos 3x O y(x)=3C,cosx+3C2sinx o y(x) =C1cos3x+C2sin3x

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dy 6. The differential equation (1+x²) - 2xy = x + x³ has the general solution dx y=e³(x+¹)+C 4 (A) (B) y = eIn(1+x²)+C(+) (C) y=x²y+C₁x² + С₂x4 (D) y = (1+x²) ln(1+x²) + C(1+x²) (E) y = Cer²(x²+x²)