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From the given equation, $$\sqrt{1+x}+\sqrt{1-x}=\frac{7\sqrt{2}}{5}.$$Squaring both sides, we get $$1+x+2\sqrt{1-x^2}+1-x=\frac{98}{25},$$which simplifies to $$2\sqrt{1-x^2}=\frac{48}{25}.$$Dividing both sides by 2, we get $$\sqrt{1-x^2}=\frac{24}{25}.$$Squaring both sides again, we get $$1-x^2=\frac{576}{625},$$so $$x^2=\frac{49}{625}.$$The positive value of $x$ is then $\boxed{\frac{7}{25}}.$