Differential Equations Answers

Solve.
d²y/dx² - 4x dy/dx + (4x²-1)y = -3exp(x²)sin(2x)

Solve the PDE.

x² d²z/dx² ⁻y² d²z/dy² + x dz/dx - y dz/dy = ln x

Find the general solution of the differential equation .
y²/2 + 2yexp(t) + (y+exp(t))dy/dt = 0

Find the orthogonal trajectories of the family of parabolas x=cy² and sketch their graph.

If y₁ = 2x+2 and y₂= -x²/2 are the solutions of the equation y= xy´+( y´)²/2 ,
Then are the constant multiples c₁y₁ and c₂y₂ , where c₁,c₂ are arbitary, also the solutions of the given DE ? Is the sum y₁+y₂ a solution ? Justify your answer .

Solve the following differential equations.

(i) dx/y²(x-y) =dy/x²(x-y) =dz/z(x²+y²)

(ii) (yz+z²)dx - xzdy + xydz =0

(iii) (z+z³)cosx dx - (z+z³)dy + (1-z²) (y-sinx)dz =0

Show that , for the DE
d²y/dx² + a(x) dy/dx+b(x)y=0

exp(mx) is a particular integral if m²+am+b=0 . Hence find the value of m so that exp(mx) is a particular integral of the equation
(x-2) d²y/dx² - (4x-7)dy/dx+(4x-6)y=0

Reduce the equation x²(y-px)=yp² to clairaut´s form and hence find its complete solution.

Find the general solution of the DE
y² lny = x py+p².
Does the equation has any singular solution? If yes, obtain it.

y'''+y''=8x^2