Question

Solve this IVP Uxx=4xy+e^x ,u(0,y),Ux(0,y)=1

Answer 1

Answer on Question #82974 – Math – Differential Equations

Question

Solve this IVP $U_{xx} = 4xy + e^x x$, $u(0,y), Ux(0,y) = 1$

Solution

$$U_x = \int U_{xx} dx = \int (4xy + e^x) dx = \int 4xy dx + \int e^x dx = 4y \frac{x^2}{2} + e^x + c$$ $$U_x(0,y) = 4y \frac{0^2}{2} + e^0 + c = 1 + c = 1 =>$$

C = 0

U = \int U_x dx = \int \left(4y \frac{x^2}{2} + e^x\right) dx = \int 4y \frac{x^2}{2} dx + \int e^x dx = 2y \frac{x^3}{3} + e^x + c

U(0,y) = 2y \frac{0^3}{3} + e^0 + c = 1 + c

\text{If } U(0,y) = 1, \text{ then}

U(0,y) = 2y \frac{0^3}{3} + e^0 + c = 1 + c = 1 =>

C = 0

=> U = 2y \frac{x^3}{3} + e^x

$$Answer:$$

U(0,y) = 1 + c

If $U(0,y) = 1$, then

U = 2y \frac{x^3}{3} + e^x.

$$

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