Question

Solve the following DEs
(i) [(dy/dx)-1]^2 [(d^2y/dx^2)+1]^2 y= sin^2 (x/2) +e^x +x
(ii) 2x^2y (d^2y/dx^2) + 4y^2 = x^2 (dy/dx)^2 + 2xy (dy/dx)

Answer 1

Answer on Question #66664 – Math – Differential Equations

Question

Solve the following DEs

i)

$$\left(\frac{dy}{dx} - 1\right)^2 \left(\frac{d^2y}{dx^2} + 1\right)^2 y = \left(\sin \frac{x}{2}\right)^2 + e^x + x$$

Solution

It is impossible to solve using analytical methods, and for numerical methods we have not initial value.

Question

ii)

$$2x^2 y \frac{d^2y}{dx^2} + 4y^2 = x^2 \left(\frac{dy}{dx}\right)^2 + 2xy \frac{dy}{dx}$$

Solution

$$u = \frac{y_x'}{y}$$ $$u_x' + \frac{1}{2}u^2 - \frac{1}{x}u + \frac{2}{x^2} = 0$$ $$v(x) = e^{\frac{1}{2} \int u(x) \, dx}$$ $$x^2 v_{xx}'' - x v_x' + v = 0$$ $$v(x) = |x| \left(C_1 + C_2 \ln |x|\right)$$ $$\ln v = \frac{1}{2} \int u(x) \, dx$$ $$\frac{v_x'}{v} = \frac{u}{2}$$ $$v _ {x} ^ {\prime} = C _ {1} + C _ {2} + C _ {2} \ln x$$ $$\frac {C _ {1} + C _ {2} + C _ {2} \ln x}{x (C _ {1} + C _ {2} \ln x)} = \frac {y _ {x} ^ {\prime}}{2 y}$$ $$\frac {C _ {1} + C _ {2} + C _ {2} \ln x}{x (C _ {1} + C _ {2} \ln x)} d x = \frac {d y}{2 y}$$ $$\int \frac {C _ {1} + C _ {2} + C _ {2} \ln x}{x (C _ {1} + C _ {2} \ln x)} d x = \int \frac {d y}{2 y}$$

Answer: $y = c_{2}x^{2}(c_{1} + \ln x)^{2}$.

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